A selective review of the question of how repulsive electron correlations might give rise to off-diagonal long-range order (ODLRO) in high-temperature superconductors is presented. The article makes detailed explanations of the relevance to superconductivity of reduced electronic density matrices and how these can be used to understand whether ODLRO might arise from Coulombic repulsions in strongly correlated electronic systems. Time-reversed electron pairs on alternant Cuprate and the iron-based pnictide and chalcogenide lattices may have a weak long-range attractive tail and much stronger short-range repulsive Coulomb interaction. The long-range attractive tail may find its origin in one of the many suggested proposals for high-T_{c} superconductivity and thus has an uncertain origin. A phenomenological Hamiltonian is invoked whose model parameters are obtained by fitting to experimental data. A detailed summary is given of the arguments that such interacting electrons can cooperate to produce a superconducting state in which time-reversed pairs of electrons effectively avoid the repulsive hard-core of the Coulomb interaction but reside on average in the attractive well of the long-range potential. Thus, the pairing of electrons itself provides an enhanced screening mechanism. The alternant lattice structure is the key to achieving robust high-temperature superconductivity with dx^{2}-y^{2} or sign alternating s-wave or s± condensate symmetries in cuprates and iron-based compounds. Some attention is also given to the question first raised by Leggett as to where the Coulombic energy is saved in the superconducting transition in cuprates. A mean-field-type model in which the condensate density serves as an order parameter is discussed. Many of the observed trends in the thermal properties of cuprate superconductors are reproduced giving strong support for the proposed model for high-temperature superconductivity in such strongly correlated electronic systems. © 2015 Wiley Periodicals, Inc.

The quest to obtain an understanding of the “electronic” mechanism of high-temperature superconductivity in strongly correlated systems such as the cuprates has been one of the main goals of condensed matter science for nearly three decades. A widely held view is that hole-doped cuprates have a condensate wave function with a d-wave symmetry shown above. This review discusses how this might arise in strongly correlated alternant systems and presents predictions for some selected thermal properties.

Quantum chemistry is an important area of application for quantum computation. In particular, quantum algorithms applied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy of atoms and molecules. The Bravyi–Kitaev transformation is a method of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian of *n* interacting fermions to an
-local Hamiltonian of *n* qubits. This is an improvement in locality over the Jordan–Wigner transformation, which results in an *O*(*n*)-local qubit Hamiltonian. We present the Bravyi–Kitaev transformation in detail, introducing the sets of qubits which must be acted on to change occupancy and parity of states in the occupation number basis. We give recursive definitions of these sets and of the transformation and inverse transformation matrices, which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan–Wigner and Bravyi–Kitaev Hamiltonians for the quantum simulation of methane using the STO-6G basis. © 2015 Wiley Periodicals, Inc.

Quantum chemistry is a promising application of quantum computation. To calculate the energy of electrons in molecules exactly and efficiently is possible on future quantum computers, but not using current conventional methods. The quantum algorithms developed for this purpose include new methods for handling the antisymmetry of the electron wavefunction. This article discusses one such method in detail together with the results of its application to the simulation of methane.

A complete electro-nuclear (EN) basis set and quantum electrodynamics bases in photon number scheme combines to form photonic bases sets. The EN *q*-states can hence be modulated by appropriate external electromagnetic sources. Quantum determinants for HCN/CNH isomerization within photonic bases are elaborated that rationalize quantum state changes as if it were an apparent unimolecular process. Topologic label of base states permit linking with those obtained with semiclassic schemes. A comparison of results leads to conclude that both schemes can turn out to be complementary. The *q*-scheme yielding more detailed information that the semiclassic one as expected. © 2015 Wiley Periodicals, Inc.

Electromagnetic radiation probes any electronuclear quantum (*q*-) state. Different frequencies produce particular *q*-states views. Atto-second pulses generate packets with very high frequency. The interaction of radiation with matter generates coherent states and a host of new frequencies are shinned back carrying information on the probed *q*-state. Pushing probes frequencies to resonate with a spectral region covered by the transition state in a chemical reaction will yield definite influence on said reaction via transient *q*-transition state structures.

The study on the absorption of toxic gases such as mustard gas by organic host is essential to the development of inexpensive detection and decontamination equipments. Using quantum chemical methods, we propose cucurbituril as an effective host to capture mustard gas. It was found that stable complexes are formed with the inclusion of the toxic gas molecules inside the cucurbituril cavity, compared with the lateral and exterior interactions. Oxygen mustard has a comparable binding energy with sulfur mustard and hence can be used during experimental investigation. Additionally, during the inclusion complex formation, the presence of heteroatoms helps the guest molecules to undergo a larger structural reorganization to get accommodated inside the cucurbituril macromolecule. Atoms-in-molecules analysis shows the existence of strong intermolecular CH…O bonding between the guest molecules and cucurbituril macromolecule. The presence of an intramolecular CH…Cl type of bonding accounts for the higher stabilization of sulfur mustard inside the cucurbituril macromolecule. © 2015 Wiley Periodicals, Inc.

Macrocyclic host molecules are successfully used to sense and encapsulate gas molecules, even at low concentration and in near room temperature. Organic supramolecules, for example, Cucurbit[n]urils, are of particular interest for this role. Density functional theory calculations provide insight into the binding ability of toxic mustard gas on cucurbituril, and suggest that oxygen-mustard is an effective model system to experimentally study the absorption properties of the much more toxic sulfur-mustard gas.

Geminals are counterparts of two-electron chemical bonds and lone pairs in the realm of wave functions. Antisymmetrized products of geminals provide a solid framework for studies of electron pairing in molecular systems. Natural advantages of geminal wave functions such as the correct description of bond breaking and formation make them a powerful model for electronic structure calculations. The concerted efforts to develop geminal-based methods concentrate on increasing their accuracy and universality. The downside is the high computational cost limiting applications to small and toy systems. In contrast, the inherent local structure of optimal geminals opens pathways to development of methods for large systems. In particular, perspectives for linear scaling and hybrid quantum/classical approaches using geminals are discussed. The same principles are proposed for development of computational schemes for covalently bound and molecular crystals. © 2015 Wiley Periodicals, Inc.

Multiconfigurational methods based on geminals effectively account pair electron correlations. The cost of their applications to large systems is prohibitive due to the complexity of geminal variation. Nevertheless, the intrinsic local character of geminals prompts development of fast methods for electronic structure calculations. In particular, linear scaling and hybrid quantum mechanical/molecular mechanical schemes as well as efficient methods for crystals are within reach.

Hydride-transfer reactions between benzylic substrates and 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) were investigated by DFT (density functional theory) calculations. The lowest unoccupied molecular orbital of DDQ has the largest extension on two carbonyl oxygens, which comes from two-step mixing of antisymmetric orbitals of fragment π MOs. Transition-state (TS) geometries and activation energies of reactions of four benzylic substrates R^{2}CH_{2}-*para*-C_{6}H_{4}R^{1} (R^{1}, R^{2} = H and/or OCH_{3}) with DDQ were calculated. M06-2X/6-311(+*)G* was found to be a practical computational method, giving energies and geometries similar to those of M06-2X/6-311++G(3df,2pd) and wB97xD/6-311++G(3df,2pd). For toluene (R^{1} = R^{2} = H), an initiation-propagation model was suggested, and the calculated kinetic isotope effect k(H)/k(D) = 5.0 with the tunnel correction at the propagating step is in good agreement with the experimental value 5.2. A reaction of *para*-MeOC_{6}H_{4}CH_{2}(OMe) + DDQ + (H_{2}O)_{14} *para*-MeOC_{6}H_{4}C(O)H + HOMe + DDQH_{2} + (H_{2}O)_{13} was investigated by M06-2X/6-311(+*)G*. Four elementary processes were found and the hydride transfer (TS1) is the rate-determining step. The hydride transfer was promoted by association with the water cluster. The size of the water cluster, (H_{2}O)* _{n}*, at TS1 was examined. Three models of

2,3-Dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) is widely used as dehydrogenation agent of hydroaromatic compounds. The result of Density Functional Theory simulations propose that a chain-propagation mechanism for the toluene + DDQ Ph-CH_{2}-DDQ-H) reaction. This mechanism reproduces satisfactorily the kinetic isotope effect in this reaction. Theory also shines a light onto the reaction paths of related reactions.

The photodetachment of hydrogen negative ion
near different inelastic surfaces is investigated by the semiclassical closed orbit theory for arbitrary laser polarization direction
. A two-term formula of photodetachment cross section consisting of a smooth background term and an oscillatory term is derived. The oscillatory term contains an extra angular factor
that describes the dependence of oscillations in total cross section on the laser polarization direction. It is observed that the amplitude of oscillations in cross section reaches maximum at
when laser polarization is parallel to the *z*-axis and it approaches zero as the laser polarization direction becomes perpendicular to the *z*-axis. It is also observed that as the reflection coefficient
, which accounts for the inelastic behavior of the surfaces, increases the amplitude of oscillation also increases. © 2015 Wiley Periodicals, Inc.

The photodetachment process of negative ions near surfaces is an innovative theoretical and experimental development. From the photodetached-electron spectrum, important properties of the ions (e.g., binding energy) and the surfaces (e.g., band structure) can be obtained. This article investigates photodetachment of hydrogen negative ion near inelastic surfaces for arbitrary laser polarization direction by semiclassical closed orbit theory.

The tridiagonal J-matrix approach has been used to calculate the low and moderately high-lying eigenvalues of the rotating shifted Tietz–Hua (RSTH) oscillator potential. The radial Schrödinger equation is solved efficiently by means of the diagonalization of the full Hamiltonian matrix, with the Laguerre or oscillator basis. Ro–vibrational bound state energies for 11 diatomic systems, namely
,
,
, NO, CO,
,
,
,
,
, and NO^{+}, are calculated with high accuracy. Some of the energy states for molecules are reported here for the first time. The results of the last four molecules have been introduced for the first time using the oscillator basis. Higher accuracy is achieved by calculating the energy corresponding to the poles of the S-matrix in the complex energy plane using the J-matrix method. Furthermore, the bound states and the resonance energies for the newly proposed inverted Tietz–Hua IRSTH-potential are calculated for the H_{2}-molecule with scaled depth. A detailed analysis of variation of eigenvalues with *n,*
quantum numbers is made. Results are compared with literature data, wherever possible. © 2015 Wiley Periodicals, Inc.

The tridiagonal J-matrix method has been successfully used for a number of situations of physical and chemical interest. Here that J-matrix method is used to calculate the Ro–vibrational eigenvalues of the Tietz–Hua (TH) potential for 11 diatomic systems, four of which are reported for the first time. The bound and resonance eigenvalues for the newly proposed inverted TH potential are calculated for the H_{2}-molecule, with scaled depth, and compared with the inverted Morse potential.

This work describes a new approach for approximate obtaining the positively defined electronic kinetic energy density (KED) from electron density. KED is presented as a sum of the Weizsäcker KED, which is calculated in terms of electron density exactly, and unknown Pauli KED. The latter is presented via local Pauli potential and Gritsenko–van Leeuwen–Baerends kinetic response potential, to which the second-order gradient expansion is applied. The resulting expression for KED contains only one empirical parameter. The approach allowed to correctly reproduce all the features of KED, and electron localization descriptors as electron localization function and phase-space defined Fisher information density for main types of bonds in molecules and molecular crystals. It is also demonstrated that the method is immediately applicable to derivation of mentioned bonding descriptors from experimental electron density. Herewith the method is significantly free from the drawback of Kirzhnits approximation, which is now commonly accepted for evaluation of the electronic kinetic energy characteristics from precise X-ray diffraction experiment. © 2015 Wiley Periodicals, Inc.

The kinetic energy density (KED) can be extracted from the electron density using local Pauli potential and second-order gradient expansion scheme. This approach is able to correctly reproduce all the features of KED and electron localization descriptors as electron localization function and the phase-space defined Fisher information density for many types of bonds in molecules and crystals. This method can be applied for the derivation of bonding descriptors from experimental electron density.

We review the spin radical pair mechanism which is a promising explanation of avian navigation. This mechanism is based on the dependence of product yields on (1) the hyperfine interaction involving electron spins and neighboring nuclear spins and (2) the intensity and orientation of the geomagnetic field. This review describes the general scheme of chemical reactions involving radical pairs generated from singlet and triplet precursors; the spin dynamics of the radical pairs; and the magnetic field dependence of product yields caused by the radical pair mechanism. The main part of the review includes a description of the chemical compass in birds. We review: the general properties of the avian compass; the basic scheme of the radical pair mechanism; the reaction kinetics in cryptochrome; quantum coherence and entanglement in the avian compass; and the effects of noise. We believe that the quantum avian compass can play an important role in avian navigation and can also provide the foundation for a new generation of sensitive and selective magnetic-sensing nano-devices. © 2015 Wiley Periodicals, Inc.

The navigational ability of birds has been a subject of interest for centuries. The radical pair mechanism of avian compass is a promising hypothesis to explain the way in which birds use the geomagnetic field to navigate themselves. Entanglement last long enough in this mechanism and can play an important role in it. The study of radical pair mechanism can also provide the foundation for a new generation of sensitive and selective magnetic-sensing nano-devices.

Electronic structure calculations are widely and increasingly utilized for understanding, designing, screening, and analyzing the material properties in various applications. Especially, for the last two decades, researches on the rechargeable battery have grown rapidly with the help of computational materials science. In this perspective, we briefly overview the current status of such progresses in the battery electrode. In particular, the configuration problem to determine the ground structure for estimating thermodynamic properties such as voltage and intermediate phase is discussed, followed by theoretical interpretations on the phase behavior and phase boundary migration. Some future challenges are commented. © 2015 Wiley Periodicals, Inc.

Some key properties of battery materials, such as voltage, intermediate phase, ion ordering, and phase boundary migration are interpreted from the computational viewpoint, providing the perspectives and challenges on the use of electronic structure calculations for battery materials science.

The source of unoccupied Ti 3*d* states in the case of stoichiometric anatase structured (TiO_{2})* _{n}* clusters has been investigated using

Computational modeling results show that certain stoichiometric anatase structured titanium dioxide clusters can have unoccupied defect states. In the classical cases, occupied defect states can be explained with titanium and oxygen surface defects, but unoccupied states have been unexplained. The origin of these empty gap states is related to effective subcluster formation which gives rise to empty defect-like gap states.

In this work, the energy gaps (*E*_{g}) of highest occupied orbitals and lowest unoccupied orbitals, trap energies (*E*_{t}(*e*) and *E*_{t}(*h*)) and excited energies of polyethylene model compound, typical defect compound, acetophonene, and 33 designed additives are obtained using density functional method at B3LYP/6–311+G(d, p) level. The correlation between trapping-electrons (holes) abilities of additives and molecular frontier orbitals is established, and a new understanding for trap mechanism based on chemical molecular orbital levels is given for the first time which could be used to filter qualitative additives as voltage stabilizers of polyethylene. The role of trap energies and the energy gaps on discussing space charge accumulation and electric breakdown is analyzed in detail. A molecular design strategy for potential additives of cross-linked polyethylene insulated high-voltage cable is shown based on conjugation effect, substituents character, and polycyclic aromatic compounds. © 2015 Wiley Periodicals, Inc.

Theoretical calculations contribute to a new understanding for the trap mechanism in polyethylene insulators based on chemical molecular orbital levels. The role of trap energies and the energy gaps on the mechanisms of space charge accumulation and electric break are explained and can result useful in the design strategy of new additives for cross-linked polyethylene insulation cable.

Potential applicability of undoped, B-, and N-doped carbon nanotubes (CNTs) for elaboration of the working materials of gas sensors of hydrogen halide molecules HX (X = F, Cl, Br) is analyzed in computational studies of molecular adsorption on the CNTs surfaces. Density Functional Theory (DFT)-based geometry-optimized calculations of the electronic structure of undoped, B-, and N-doped CNTs of (3,3) and (5,5) chiralities with adsorbed HX (X = F, Cl, Br) molecules are performed within molecular cluster approach. Relaxed geometries, binding energies between the adsorbates and the nanotubes, charge states of the adsorbates and the electronic wave function contours are calculated and analyzed in the context of gas sensing applications. Obtained results are supplemented by calculations of adsorption of hydrogen halides on B(N)-doped graphene sheets which are considered as model approximation for large-diameter CNTs. It is found that the B-doped CNTs are perspective for elaboration of sensing materials for detection of HCl and HBr molecules. The undoped and the N-doped CNTs are predicted to be less suitable materials for detection of hydrogen halide gases HX (X = F, Cl, Br). © 2015 Wiley Periodicals, Inc.

Widely present as regent or bioproducts in industrial processes, hydrogen halide gases HX (X = F, Cl, Br) are very toxic. Undoped and B- and N-doped carbon nanotubes (CNTs) are tested as working materials of gas sensors of hydrogen halides. Density functional theory calculations suggest that B-doped CNTs are promising materials for sensing HCl and HBr gases. Undoped and N-doped CNTs are found to be less promising materials for sensing of hydrogen halides.

The seven rhenium (I) tricarbonyl complexes having a general formula *fac*-[ReBr(CO)_{3}(R_{1},R_{2},R_{3}-N^{^}N)] (N^{^}N = imidazo[4,5-f]-1,10-phenanthroline; R_{1} = ^{t}Bu, R_{2} = R_{3} = H, **1**; R_{1} = CCH, R_{2} = R_{3} = H, **2**; R_{1} = ^{t}Bu, R_{2} = CCH, R_{3} = H, **3**; R_{1} = ^{t}Bu, R_{2} = R_{3} = CCH, **4**; R_{1} = ^{t}Bu, R_{2} = CH_{3}, R_{3} = H, **5**; R_{1} = ^{t}Bu, R_{2} = R_{3} = CH_{3}, **6**; R_{1} = ^{t}Bu, R_{2} = OCH_{3}, R_{3} = H, **7**) have been investigated theoretically by density functional theory (DFT) and time-dependent density functional theory (TDDFT) methods. The different substituted groups on N^{^}N ligand induce changes on the electronic structures and photophysical properties for these complexes. It is found that the introduction of CC decreases the energy level of lowest unoccupied molecular orbital (LUMO) while the introduction of CH_{3} or OCH_{3} lead to increase the energy level of LUMO. The order of LUMO energy level rising is in line with the increasing of donating abilities of substituted groups; and the influence of R_{2} position is greater than that of R_{1} position on LUMO energy level. The lowest energy absorption bands have changes in the order of **7** < **6** < **5** < **1** < **2** < **3** < **4**. These results of electronic affinity (EA), ionization potential (IP), and reorganization energy (*λ*) indicate that all of these complexes can be used as electron transporting materials. Moreover, the smallest difference between *λ*_{electron} and *λ*_{hole} of **4** indicates that it is better to be used as an emitter in the organic light-emitting diodes. © 2015 Wiley Periodicals, Inc.

Rhenium-tricarbonyl chromophoric complexes are designed so that their fluorescence behaviors can be tuned by the use of different substitute groups. First-principles modeling is used to understand the relationship between different ancillary ligands and properties, photophysics of these rhenium (I) tricarbonyl complexes. Modeling can lead the way in developing these materials for applications in OLEDs.

Effect of external electric field on interaction energy as well as stability of the hydrogen-bonding, stacking, and OH
π*-*bonded systems are analyzed in the light of density functional theory (DFT) and conceptual DFT. Interaction energy and stability measured in terms of global hardness and highest occupied molecular orbital energy of the considered systems are observed to be sensitive toward the strength and direction of the applied external electric field. The curvature of the potential energy surfaces gets changed in presence of an external electric field. © 2015 Wiley Periodicals, Inc.

Interaction energy in hydrogen-bonding, stacking, and XH π-bonded systems show remarkable response toward the strength and direction of the applied electric field.

Theoretical investigation on the gas-phase degradation reaction mechanism of methamidophos (MAP) and chloramine phosphorus (CHP) with OH radicals is performed. The equilibrium geometries and the harmonic vibration frequencies of the stationary points are obtained at M06-2x/6-31+G(d,p) level, and the higher-level energetic information is further refined at M06-2x/6–311++G(3df,2p) level. The rate constants for the 14 reaction channels are calculated by the improved canonical variational transition state theory with small-curvature tunneling correction over the temperature range 200–2000 K. The three-parameter expressions of *k*_{1}(*T*) = 1.53 × 10^{−19}*T*^{2.74}exp(−1005.12/*T*), *k*_{2}(*T*) = 1.36 × 10^{−20}*T*^{3.02}exp(−1259.56/*T*) are given. The total rate constants of all reaction channels of MAP with OH radicals are in good agreement with the available experimental data. Our results indicate that the H-abstraction reactions on methyl are the major channels for the reaction of MAP and CHP with OH radicals. © 2015 Wiley Periodicals, Inc.

Methamidophos and chloramine phosphorus are organophosphate pesticides in common use in agriculture. They can be eliminated in the atmosphere by degradation reactions with OH and NO_{3} radicals and ozone. First-principle modeling can provide a fundamental insight into the reaction mechanisms of degradation of these species by OH radical. Simulation results suggest that the H-abstraction steps on methyl are the major channels for these reactions.

The dynamics and mechanism of proton exchange in phosphonic acid-functionalized polymers were studied using poly(vinyl-phosphonic acid) (PVPA) as a model system along with quantum chemical calculations and Born–Oppenheimer molecular dynamics (BOMD) simulations at the B3LYP/TZVP level as model calculations. This theoretical study began with searching for the smallest, most active polymer segments and their intermediate conformations which could be involved in the local proton-exchange process. The B3LYP/TZVP results confirmed that a low local dielectric environment and excess proton conditions are required to generate the intermediate conformations, and the shapes of the potential energy curves of the proton exchange between the two phosphonic acid functional groups are sensitive to the local conformational changes. In contrast, a high local dielectric environment increases the energy barriers, thereby preventing the proton from returning to the original functional group. Based on the static results, a mechanism for the proton exchange between the two functional groups involving fluctuations in the local dielectric environment and a local conformational change was proposed. The BOMD results confirmed the proposed mechanism by showing that the activation energies for the proton exchange in the hydrogen bond between two immobilized phosphonic acid moieties, obtained from the exponential relaxation behaviors of the envelopes of the velocity autocorrelation functions and the ^{1}H Nuclear Magnetic Resonance (NMR) line-shape analyses, are too low to be the rate-determining process. Instead, coupled librational motion in the backbone which leads to the interconversion between the two intermediate conformations possesses higher activation energy, and therefore represents one of the most important rate-determining processes. These findings suggested that the rate of the proton exchange in the model phosphonic acid-functionalized polymer is determined by the polymer mobility which, in this case, is the large-amplitude librational motion of the vinyl backbone. © 2015 Wiley Periodicals, Inc.

Because of their high proton conductivity and mechanical strength, phosphonic acid-functionalized polymers are candidates for the electrolyte membranes in high-temperature FC. Low local-dielectric environment and excess proton conditions are required to generate the intermediate conformations involved the local proton-exchange process. The rate of the proton exchange in the model phosphonic acid-functionalized polymer is found to be determined by the polymer mobility which, in this case, is the large-amplitude librational motion of the vinyl backbone.

A quantum chemical investigation on the reaction mechanism of CH_{3}O_{2} with OH has been performed. Based on B3LYP and QCISD(T) calculations, seven possible singlet pathways and seven possible triplet pathways have been found. On the singlet potential energy surface (PES), the most favorable channel starts with a barrierless addition of O atom to CH_{3}O_{2} leading to CH_{3}OOOH and then the OO bond dissociates to give out CH_{3}O + HO_{2}. On the triplet PES, the calculations indicate that the dominant products should be ^{3}CH_{2}O_{2} + H_{2}O with an energy barrier of 29.95 kJ/mol. The results obtained in this work enrich the theoretical information of the title reaction and provide guidance for analogous atmospheric chemistry reactions. © 2015 Wiley Periodicals, Inc.

Methyl peroxy radical (CH_{3}O_{2}) plays an important role in the photochemical cycling of ozone in the atmosphere. Its reaction with OH radical is one of the most important degradation processes in atmospheric chemistry. Electronic structure calculations are helpful to explore the reaction pathway for the CH_{3}O_{2} + radical OH and analogous reactions.

The rise of quantum information science has opened up a new venue for applications of the geometric phase (GP), as well as triggered new insights into its physical, mathematical, and conceptual nature. Here, we review this development by focusing on three main themes: the use of GPs to perform robust quantum computation, the development of GP concepts for mixed quantum states, and the discovery of a new type of topological phases for entangled quantum systems. We delineate the theoretical development as well as describe recent experiments related to GPs in the context of quantum information. © 2015 Wiley Periodicals, Inc.

A quantum state taken around a loop in state space may pick up a phase factor of geometric origin. This phase factor can be used to implement robust quantum gates that process information in a quantum computer. Conversely, tools in quantum information science can be used to obtain new insights into physical, mathematical, and conceptual nature of the geometric phase.

The treatment of high-dimensional problems such as the Schrödinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods—developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools—can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic “*New wavefunction methods and entanglement optimizations in quantum chemistry*” at the Workshop on Theoretical Chemistry, February 18–21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.

Building on research performed in the past decade, entanglement-based methods,—developed for different purposes in distinct communities—have matured into tools that can be employed to attack highly challenging problems in quantum chemistry. For example, controlled manipulation of quantum entanglement in tensor product approximations yields a novel approach to treat high-dimensional optimization problems and time-dependent equations, such as the Schrödinger equation.

Using density functional theory calculations, the adsorption and catalytic decomposition of formic acid (HCOOH) over Si-doped graphene are investigated. For the stable adsorption geometries of HCOOH over Si-doped graphene, the electronic structure properties are analyzed by adsorption energy, density of states, and charge density difference. A comparison of the reaction pathways reveals that both dehydration and dehydrogenation of HCOOH can occur over Si-doped graphene. The estimated reaction energies and the activation barriers suggest that for the dehydration of HCOOH on the Si-doped graphene, the rate-controlling step is H + OH H_{2}O reaction. For the dehydrogenation of HCOOH, the rate-determining step is the breaking of the CH bond of the HCOO group to form the CO_{2} molecule and the atomic H. Our results reveal that the low cost Si-doped graphene can be used as an efficient nonmetal catalyst for OH bond cleavage of HCOOH. © 2015 Wiley Periodicals, Inc.

The catalytic decomposition reaction of formic acid (HCOOH) is considered an important step for the generation of pure hydrogen in electric fuel cells. Si-doped graphene provides an alternative to more traditional catalysts for the decomposition of HCOOH, based on noble metals. Density functional theory calculations explore both the dehydration and dehydrogenation pathways of HCOOH on this efficient nonmetal catalyst.

Several anion-π complexes of isocyanuric acid, thioderivatives and their halogen substituted derivatives with chloride anion have been studied. The geometric and energetic features, charges transfer from chloride anion to the aromatic rings and “atoms-in-molecules” analysis are performed and discussed for these complexes. The results show that the strength of the anion-π interaction between cyanuric derivatives and chloride anion can be tuned by halogen-substituting. The localized molecular orbital energy decomposition analyses shows that, in the total interaction, exchange and electrostatic energies are the dominant stabilizing forces, and the polarization energies also make a favorable contribution. Finally, solvent effect significantly weakens the anion-π interaction between the isocyanuric derivatives and chloride anion, especially in polar solvents. © 2015 Wiley Periodicals, Inc.

The interactions between π-electron-deficient aromatic rings and anions have been recognized possessing pivotal role in many key chemical and biological processes, especially with respect to molecular recognition. Therefore, the design of highly selective anion receptor has attracted considerable attention in recent years. The anion-π interaction can be tuned via halogen substituent effects in cyanuric acids and its derivatives. This principle may be applicable to the design of new neutral anion receptors.

The Pauli exclusion principle requires that the occupations of the orbitals lie between zero and one. These Pauli conditions hold for one-electron reduced density matrices (1-RDMs) from both open and closed quantum systems. More than 40 years ago, it was recognized that there are additional conditions on the 1-RDM for closed quantum systems. In this review, we discuss the structure of the 1-RDM from the generalized Pauli exclusion principle in many-electron atoms and molecules and the violation of the generalized Pauli principle as a sufficient condition for the openness of a many-electron quantum system. © 2015 Wiley Periodicals, Inc.

The Pauli exclusion principle requires that the occupations of the orbitals lie between zero and one. These Pauli conditions hold for one-electron reduced density matrices (1-RDMs) from both open and closed quantum systems. More than 40 years ago, it was recognized that there are additional conditions on the 1-RDM for closed quantum systems. The violation of the generalized Pauli principle is discussed as a sufficient condition for the openness of a many-electron quantum system.

Quantum entanglement features exhibited by the reaction path of some selected elementary chemical reactions: hydrogenic abstraction, nucleophilic hydrogenic substitution, three-atom insertion reaction of silylene into hydrogen, and the cycloaddition of cyclopentadiene into anhydride maleic are investigated in this work. The phenomenological behavior of these reactions is described by two of the fundamental descriptors of the molecular densities, the atomic charges, and the electric potentials, to associate the maximum entangled transition state (METS) to the concurrent processes of the chemical reactions. It is found that the METS characterizes the transition state of symmetrical reactions; and for nonsymmetrical ones, it features a new critical point along the intrinsic reaction path. In addition, benchmark calculations of the relevant quantitative entanglement measures for the critical points of these reactions are reported. © 2015 Wiley Periodicals, Inc.

A quantum information looks at basic chemical reactions reveal the relevance of entanglement for characterizing their different stages. In particular, the maximal-entangled transition state describes the transition state of symmetrical reactions. In the case of nonsymmetrical reactions, a new critical point along the intrinsic reaction path, indicating concurrent processes of charge/potential equalization, is found.

Titania–water interfaces are important in various fields of science, from geophysics to photocatalysis and biochemistry. Here, we use *ab initio* molecular dynamics simulations to investigate the structure of thin water overlayers on the (101) surface of TiO_{2} anatase in the presence of oxidizing defects. For comparison, results of our previous studies of water layers on defect-free and reduced anatase (101) are also reviewed. On the stoichiometric defect-free surface-ordered structures are formed at one and two monolayer coverage, and the order in the first bilayer is largely maintained when a third water layer is adsorbed. By contrast, the vertical and in-plane ordering of the water layers is strongly perturbed in the presence of both oxidizing and reducing defects. As a result, the structure of the water layer is much more diffuse under these conditions, and frequent exchanges of water molecules between different layers are observed. © 2015 Wiley Periodicals, Inc.

Oxide-water interfaces are important in many fields of science and TiO_{2}-water interfaces are of special interest due to the many applications of this material. This article discusses the structure of thin water layers on defect-free and defective anatase, the TiO_{2} polymorph more relevant in photocatalysis, on the basis of ab initio molecular dynamics simulations. The results show that surface defects significantly affect the structure and dynamical properties of the adsorbed water layers at the interface.

Intersystem crossing is a radiationless process that can take place in a molecule irradiated by UV-Vis light, thereby playing an important role in many environmental, biological and technological processes. This paper reviews different methods to describe intersystem crossing dynamics, paying attention to semiclassical trajectory theories, which are especially interesting because they can be applied to large systems with many degrees of freedom. In particular, a general trajectory surface hopping methodology recently developed by the authors, which is able to include nonadiabatic and spin-orbit couplings in excited-state dynamics simulations, is explained in detail. This method, termed SHARC, can in principle include any arbitrary coupling, what makes it generally applicable to photophysical and photochemical problems, also those including explicit laser fields. A step-by-step derivation of the main equations of motion employed in surface hopping based on the fewest-switches method of Tully, adapted for the inclusion of spin-orbit interactions, is provided. Special emphasis is put on describing the different possible choices of the electronic bases in which spin-orbit can be included in surface hopping, highlighting the advantages and inconsistencies of the different approaches. © 2015 Wiley Periodicals, Inc.

Intersystem crossing mediated by spin-orbit coupling provides, together with internal conversion, a way for molecules to lose energy non-radiatively. For large molecular systems, trajectory surface hopping theories have become very popular to study excited-state dynamics, but only recently, interest in surface hopping methods incorporating spin-orbit interaction couplings has emerged. This paper describes SHARC, a generally applicable and accurate trajectory surface hopping method to describe intersystem crossing dynamics.

Organic electronic materials remarkably illustrate the importance of the “weak” dispersion interactions that are neglected in the most cost-efficient electronic structure approaches. This work introduces a fast atom-pairwise dispersion correction, dDMC that is compatible with the most recent variant of self-consistent charge density functional tight binding (SCC-DFTB). The emphasis is placed on improving the description of π-π stacked motifs featuring sulfur-containing molecules that are known to be especially challenging for DFTB. Our scheme relies upon the use of Mulliken charges using minimal basis set that are readily available from the DFTB computations at no additional cost. The performance and efficiency of the dDMC correction are validated on examples targeting energies, geometries, and molecular dynamic trajectories. © 2014 Wiley Periodicals, Inc.

dDMC is a novel atom-pairwise dispersion correction that is compatible with self-consistent charge density functional tight binding (SCC-DFTB). dDMC relies upon the use of Mulliken charges that are readily available from the DFTB computations at no additional cost. The performance of dDMC is validated on series of examples targeting energies, geometries and molecular dynamic trajectories of π-π stacked motifs featuring sulfur-containing molecules that are known to be especially challenging for DFTB.

The applications of solvatochromism have enormously increased in the last years especially when combined with the use of fluorescence-based techniques. By conjugating solvatochromic fluorescent probes to biologically relevant systems, the functions, activities, and interactions of such species in the context of living systems can be studied with spatial and temporal resolution. It is therefore of large interest to have theoretical methods and computational tools able to predict the sensitivity in both absorption and fluorescence properties of molecular probes to the environment. Here, we present the current status of the multiscale strategy based on the combination of quantum chemistry with classical models. Potentials and limitations of the approach are discussed and an outlook on future developments is given with emphasis on the role that solvation dynamics can play. © 2015 Wiley Periodicals, Inc.

Solvatochromism is a very complex phenomenon which manifests the modifications induced in the electronic states of molecular systems by the environment. Nowadays, solvatochromism can be successfully simulated with multiscale approaches combining a quantum mechanical description of the chromophoric probe and classical models for the environment.

Asymmetric catalysis is essential for the synthesis of chiral compounds such as pharmaceuticals, agrochemicals, fragrances, and flavors. For rational improvement of asymmetric reactions, detailed mechanistic insights are required. The usefulness of quantum mechanical studies for understanding the stereocontrol of asymmetric reactions was first demonstrated around 40 years ago, with impressive developments since then: from single-point Hartree–Fock/STO-3G calculations on small organic molecules (1970s), to the first full reaction pathway involving a metal-complex (1980s), to the beginning of the density functional theory-area, albeit typically involving truncated models (1990s), to current state-of-the-art calculations reporting free energies of complete organometallic systems, including solvent and dispersion corrections. The combined studies show that the stereocontrol in asymmetric reactions largely is exerted by nonbonding interactions, including CH/π attraction and repulsive forces. The ability to rationalize experimental results opens up for the possibility to predict enantioselectivities or to design novel catalysts on basis of *in silico* results. © 2015 Wiley Periodicals, Inc.

Quantum mechanical methods are nowadays widely employed to study asymmetric reactions. This review presents a historic perspective, from one of the first single-point HF/STO-3G studies reported in the 1970s to current DFT approaches involving non-truncated structures, diastereomeric reaction pathways, and free energies with solvent and dispersion corrections. The reactions discussed include nucleophilic addition, amine-catalyzed aldol reactions, osmium-catalyzed dihydroxylation, and rhodium-, ruthenium-, and iridium-catalyzed hydrogenations.

Three different enzymes are discussed, cytochrome c oxidase, involved in aerobic respiration, cytochrome c dependent nitric oxide reductase, involved in denitrification (anaerobic respiration), and photosystem II, involved in photosynthesis. For all three systems, free energy profiles for the entire catalytic cycle are obtained from quantum mechanical calculations on large cluster models of the active sites, using hybrid density functional theory with the B3LYP* functional. The free energy profiles are used to solve different fundamental problems concerning energy conservation, enzymatic reaction mechanisms and structure, and also to explain experimental results that seem to be in conflict with each other. Possible future applications to related problems using similar methodology are suggested. © 2015 Wiley Periodicals, Inc.

Cellular energy conservation and energy storage processes are important in all forms of life. Quantum chemical calculations can provide fundamental insight into the mechanisms of enzymes involved in cellular energy conservation. Large molecular models of the active sites of such enzymes are used to construct free energy profiles. From the energy profiles, conclusions are drawn about mechanisms and structures, and experimental puzzles are solved.

Quantum algorithms for quantum chemistry map the dynamics of electrons in a molecule to the dynamics of a coupled spin system. To reach chemical accuracy for interesting molecules, a large number of quantum gates must be applied which implies the need for quantum error correction and fault-tolerant quantum computation. Arbitrary fault-tolerant operations can be constructed from a small, universal set of fault-tolerant operations by gate compilation. Quantum chemistry algorithms are compiled by decomposing the dynamics of the coupled spin-system using a Trotter formula, synthesizing the decomposed dynamics using Clifford operations and single-qubit rotations, and finally approximating the single-qubit rotations by a sequence of fault-tolerant single-qubit gates. Certain fault-tolerant gates rely on the preparation of specific single-qubit states referred to as magic states. As a result, gate compilation and magic state distillation are critical for solving quantum chemistry problems on a quantum computer. We review recent progress that has improved the efficiency of gate compilation and magic state distillation by orders of magnitude. © 2015 Wiley Periodicals, Inc.

Quantum chemistry algorithms on a quantum computer map the molecular Hamiltonian to a spin Hamiltonian. The dynamics of the Hamiltonian are then simulated using a set of elementary operations. Fault-tolerant protocols limit the set of operations and implementation of certain operations requires ancillary resource states, known as magic states. Recent developments that have improved the efficiency of magic state preparation and the compilation of arbitrary gates from discrete gate sets are reviewed.

It has been predicted that entanglement creation should be associated with a lowering of energy (as well as the well-known decrease of entropy) for certain types of interaction potentials. This is a principally important question and it is shown here that np-scattering has specific features that would make it suitable for experimental tests of this prediction using neutron scattering on hydrogen at high angles. For this purpose, the evolution of n-p entanglement in Compton scattering on protons at epithermal energies (20–200 eV) is calculated over the separation distance 0–1 Å, corresponding to times up to 10^{−15} s after the collision. If an energy transfer is associated with the entanglement, it ought to be observable under these measurement conditions. © 2015 Wiley Periodicals, Inc.

The possible energetic consequences of quantum entanglement are investigated, using p-n scattering as an example. It is first calculated (for the first time for a realistic case) how the entanglement arises during the interaction of the neutron and the proton and how it decays with the separation of the particles. In the entanglement process, energy is lost through virtual excitation of a compound deuteron state, resulting in a lowering of energy.

The dynamics of stretching–stretching (SS) and stretching–bending (SB) entanglement in the normal mode molecule SO_{2} are studied by the Lie algebraic method. The influence of the decoherence from the remaining vibrational mode is considered. We have found that SS vibrations can achieve a higher degree of entanglement than the SB vibrations, and the degeneration rate of SB entanglement is higher than the SS entanglement. By the comparative study, the SS vibrations are found to be more suitable to construct a biqubit system rather than the stretching–bending vibrations. © 2014 Wiley Periodicals, Inc.

The stretching–stretching and stretching–bending vibrations in the triatomic molecule SO_{2} are investigated as possible building blocks of a bipartite qubit system. The remaining stretching or bending vibrational mode is one resource of intramolecular decoherence. Because of the strong coupling between vibrational modes in SO_{2}, the decoherence process of stretching–bending vibrations is very serious. The stretching–stretching vibrational entanglement is estimated to be more robust, and thus more suitable for the construction of qubits.

The basic concepts of orbital entanglement and its application to chemistry are briefly reviewed. The calculation of orbital entanglement measures from correlated wavefunctions is discussed in terms of reduced *n*-particle density matrices. Possible simplifications in their evaluation are highlighted in case of seniority-zero wavefunctions. Specifically, orbital entanglement allows us to dissect electron correlation effects in its strong and weak contributions, to determine bond orders, to assess the quality and stability of active space calculations, to monitor chemical reactions, and to identify points along the reaction coordinate where electronic wavefunctions change drastically. Thus, orbital entanglement represents a useful and intuitive tool to interpret complex electronic wavefunctions and to facilitate a qualitative understanding of electronic structure and how it changes in chemical processes. © 2014 Wiley Periodicals, Inc.

Quantum information theory provides tools to quantify the interaction of orbitals and allows us to measure the correlation between orbital pairs beyond the qualitative picture of molecular orbital theory. Orbital entanglement offers an alternative perspective to well-established concepts in chemistry and facilitates an understanding of electronic structures and how they change in chemical processes.

Recent advances in the design and application of redox-active fluorescent proteins (FPs) stimulated an interest in the electronic structure of the ionized/electron-detached FP chromophores. Here, we report the results of a computational study of the electron-detached and ionized states of model chromophores of green and red FPs. We focus on the analysis of the effects of the phenolate OH group position (ortho, meta, and para) on relative energies of the chromophores in the ground as well as in the ionized/detached electronic states. We found that, similarly to the green chromophore, the red chromophores with the OH group in meta position have lower vertical detachment energies (DE) and greater ionization energies relative to the ortho and para forms. Moreover, the effect is stronger for the red anionic chromophores. The differences in DE in meta species relative to their para counterparts are 0.47 and 0.25 eV for the red and green chromophores, respectively. The observed trends are due to a combined effect of resonance stabilization and the electronegativity of the acylimine group in the red chromophores. The analysis is supported by the computed charge and spin density delocalization patterns. © 2014 Wiley Periodicals, Inc.

Redox-active fluorescent proteins (FPs) are significant to many bioimaging applications and protein engineering efforts. Theoretical studies reveal strong effects of the OH-group position and the size of the π- conjugated system on the relative energies of the ground and electron-detached states of the chromophores. The effects of phenolate OH-group position on vertical ionization/detachment energies, as well as spin and charge delocalization, highlight the importance of resonance stabilization and substituent electronegativity in determining oxidation energetics.

In this article, I will discuss the overlap between the concept of Shannon entropy and the concept of electronic correlation. Quantum Monte Carlo numerical results for the uniform electron gas are also presented; these latter on the one hand enhance the hypothesis of a direct link between the two concepts but on the other hand leave a series of open questions which may be used to trace a roadmap for the future research in the field. © 2014 Wiley Periodicals, Inc.

Links between the electron density of DFT and the many-body wavefunctions/correlations are put forward in terms of encoding/decoding concepts of information theory. In particular, the concept of Shannon entropy can be employed into the development of electronic correlation functionals, and the Hohenberg−Kohn theorem be seen as a specific case of a more general encoding process.

The Euler equation of the orbital-free density functional theory is formulated with the specific Shannon and Fisher information. One of the new forms contains only the specific Shannon information. In spherically symmetric systems, the Euler equation can be formalized with the specific Fisher information only. Both the Fisher information and the length of the local wave vector are descriptors of the spherically symmetric Coulomb systems. © 2014 Wiley Periodicals, Inc.

Information theory concepts are getting traction in both physics and chemistry. Very relevant for density function theory is the relationship established between the quantum mechanical kinetic energy functional and Fisher information. The Shannon information has instead proved to be a very useful tool in analyzing atoms and molecules. In this article, the Euler equation (independent from the number of electrons) of the orbital-free density functional theory is reformulated with the Shannon and Fisher information.

Quantum-generalized entropic descriptors of the complex electronic states and their information distances are reexamined and applied to the phase-equilibria in molecules. The relation between densities of the ordinary Fisher and Shannon measures of information content is used in determining their supplements due to phases/currents. These nonclassical terms complement the familiar classical (probability) functionals of information theory in the resultant information descriptors. The nonclassical Shannon entropy measures the average magnitude of the system phase distribution, while the current term in the related Fisher measure accounts for the gradient content of the state phase. The density constrained (vertical) and unconstrained (horizontal) equilibria in molecules are distinguished. The consistency requirement that the extreme entropic principles in terms of both these resultant measures have common solutions calls for the modified, negative sign of the nonclassical Fisher indeterminicity term. The equilibrium criteria are shown to give rise to the unitary phase-transformation of molecular states in a “thermodynamic” representation of quantum-mechanical description. Possible applications of this generalized description are discussed and thermodynamical analogies are commented upon. A separation of the density (modulus) and current (phase) factors of general many-electron states is effected using the Harriman–Zumbach–Maschke construction of antisymmetric states yielding the specified electron density. A phenomenological description of molecular subsystems is outlined, which accounts for both the density and phase degrees-of-freedom of electronic states, and the current promotion of molecular fragments is explored. © 2014 Wiley Periodicals, Inc.

Information-theoretic description of electronic states requires the classical (probability) descriptors and their nonclassical (current) complements. The unconstrained extremum of the resultant entropy/information content determines the phase-transformed equilibrium state, which differs from the stationary (zero-current) state of quantum mechanics. The nonclassical supplements to the familiar classical measures, global (Shannon), and local (Fisher), are designed and current promotion of molecular fragments is explored. This fully quantum information-theoretic approach generates thermodynamic perspective on time evolution of equilibrium states.

Quantum Chemical Topology (QCT) provides topological atoms for the novel protein force field QCTFF. On page 1005 (DOI:10.1002/qua.24900), an energy transferability study on oligopeptides by Paul L.A. Popelier shows that the alpha carbon in penta-alanine (center) may as well have been extracted from tri-alanine (top left). Hundreds of configurations of tri-alanine are then sampled via normal mode distortion leading to hundreds of *ab initio* wave functions. QCT then cuts out the alpha carbon from each wave function (top right) and calculates its multipole moments, and intra-atomic and inter-atomic energy components, for which Kriging models are obtained (middle right). The image was prepared with the help of Peter Maxwell.

An atomic energy contribution to the total energy in a high-dimensional neural network potential for liquid water is presented on page 1032 (DOI:10.1002/qua.24890) by Jörg Behler. The transparent blue sphere defines the local atomic environment of one of the oxygen atoms that is highlighted in yellow. Its atomic energy depends on the positions of all atoms shown as large spheres within a cutoff radius. During molecular dynamics simulations, atoms can enter and leave this local environment.

The image represents the *atomic neighbor density* around a central atom (indicated by the white spot and the arrow), which is often used as the starting point for generating rotationally and permutationally invariant descriptors of the environment of an atom. In the Gaussian Approximation Potentials scheme reported by Albert P. Bartòk and Gábor Csányi on page 1051 (DOI:10.1002/qua.24927), the neighbor density is constructed by placing a Gaussian function at the location of each atom and then convolved with a smooth cutoff function that provides compact support. This neighbor density is then expanded in spherical harmonics and a complete set of invariants are provided by the corresponding structure factors. The invariants can be used to construct a *kernel*, which measures the similarity between neighbor environments and forms the basis for all machine learning and regression methods.

In this perspective, we explain the strategy behind QCTFF, the current name for a novel atomistic protein force field. The atoms are constructed using Quantum Chemical Topology (QCT). These topological atoms determine how a system's energy is partitioned. We give a brief account of the results hitherto obtained, and a glimpse of unpublished results. Combining this QCT partitioning with the universal quantum expression of energy, leads to four types of fundamental energy contributions. The first of these is intra-atomic and the remaining three interatomic: (i) atomic self-energy, (ii) Coulomb energy, (iii) exchange energy, and (iv) correlation energy. All structural and dynamic effects emerge from the interplay of these contributions. The machine learning method kriging captures well how they change in response to a change in nuclear configuration. Second, the Coulomb energy is represented by a converging multipolar series expansion when the nuclei are sufficiently far apart. © 2015 The Authors International Journal of Quantum Chemistry Published by Wiley Periodicals, Inc.

A novel protein force called QCTFF is parameterised without using the familiar force field expression of bonded and non-bonded interactions. Instead, it overhauls this architecture and directly captures the behaviour of an atom while interacting with other atoms. The machine learning method kriging is trained to predict four fundamental energy contributions: (i) atomic self-energy, (ii) Coulomb energy, (iii) exchange energy, and (iv) correlation energy. Multipole moment represents the medium to long-range electrostatics.

Development and applications of neural network (NN)-based approaches for representing potential energy surfaces (PES) of bound and reactive molecular systems are reviewed. Specifically, it is shown that when the density of *ab initio* points is low, NNs-based potentials with multibody or multimode structure are advantageous for representing high-dimensional PESs. Importantly, with an appropriate choice of the neuron activation function, PESs in the sum-of-products form are naturally obtained, thus addressing a bottleneck problem in quantum dynamics. The use of NN committees is also analyzed and it is shown that while they are able to reduce the fitting error, the reduction is limited by the nonrandom nature of the fitting error. The approaches described here are expected to be directly applicable in other areas of science and engineering where a functional form needs to be constructed in an unbiased way from sparse data. © 2014 Wiley Periodicals, Inc.

Neural Network-based methods for potential energy surface (NN PESs) fitting can be considered “black box” in that they do not impose any predefined functional form, and thus are very portable. However, their main disadvantage is that their accuracy is guaranteed only for a sufficiently high density of fitting points. In this case, NN PESs are arguably very accurate for small molecules and as good as other fitting methods for larger molecules.

B3LYP is currently the most widely used density functional approximation, while the X1 family of methods, namely X1, X1s, and X1se, is a set of neural network-based methods that systematically correct the B3LYP errors. The performance of the X1 family of methods in the prediction of heats of formation (HOFs), bond dissociation enthalpies (BDEs), heats of isomerization (HOIs), and so forth, is summarized against some well-established benchmarking datasets. X1 significantly eliminates the notorious size-dependent errors of B3LYP in prediction of HOFs for larger molecules. X1s further exhibits a significant improvement for BDE calculations. X1se continues to improve its predecessors on HOIs. Such a progressive improvement relies on the increasingly comprehensive descriptors. Limitations of the present approaches and the direction for future improvements are discussed. © 2015 Wiley Periodicals, Inc.

B3LYP is currently one of the most widely used density functional approximations. A class of methods (presented under the collective term of X1 family) utilizes neutral networks to correct molecular heats of formation, heats of isomerization, and bond dissociation enthalpies computed with the B3LYP functional. As the X1 family of methods does not use descriptors that require additional calculations, the corrections come at no additional computational cost, as compared with standard B3LYP calculations.

A lot of progress has been made in recent years in the development of atomistic potentials using machine learning (ML) techniques. In contrast to most conventional potentials, which are based on physical approximations and simplifications to derive an analytic functional relation between the atomic configuration and the potential-energy, ML potentials rely on simple but very flexible mathematical terms without a direct physical meaning. Instead, in case of ML potentials the topology of the potential-energy surface is “learned” by adjusting a number of parameters with the aim to reproduce a set of reference electronic structure data as accurately as possible. Due to this bias-free construction, they are applicable to a wide range of systems without changes in their functional form, and a very high accuracy close to the underlying first-principles data can be obtained. Neural network potentials (NNPs), which have first been proposed about two decades ago, are an important class of ML potentials. Although the first NNPs have been restricted to small molecules with only a few degrees of freedom, they are now applicable to high-dimensional systems containing thousands of atoms, which enables addressing a variety of problems in chemistry, physics, and materials science. In this tutorial review, the basic ideas of NNPs are presented with a special focus on developing NNPs for high-dimensional condensed systems. A recipe for the construction of these potentials is given and remaining limitations of the method are discussed. © 2015 Wiley Periodicals, Inc.

In this tutorial review, a method to construct high-dimensional interatomic potentials employing artificial neural networks is reviewed. This approach allows one to carry out molecular dynamics simulations of large systems containing thousands of atoms with close to first-principles accuracy and has been applied successfully to a number of different systems including metals, semiconductors, oxides, and molecular clusters. A strong focus of the review is on practical aspects of constructing these potentials.

We present a swift walk-through of our recent work that uses machine learning to fit interatomic potentials based on quantum mechanical data. We describe our Gaussian approximation potentials (GAP) framework, discuss a variety of descriptors, how to train the model on total energies and derivatives, and the simultaneous use of multiple models of different complexity. We also show a small example using QUIP, the software sandbox implementation of GAP that is available for noncommercial use. © 2015 Wiley Periodicals, Inc.

Interatomic potentials based on first principles data can be generated using machine learning methods. The Gaussian Approximation Potential framework puts this concept into practice, and its software implementation, QUIP, has been made available. QUIP might be used as a standalone tool or it can be easily interfaced with mainstream molecular simulation packages.

Models that combine quantum mechanics (QM) with machine learning (ML) promise to deliver the accuracy of QM at the speed of ML. This hands-on tutorial introduces the reader to QM/ML models based on kernel learning, an elegant, systematically nonlinear form of ML. Pseudocode and a reference implementation are provided, enabling the reader to reproduce results from recent publications where atomization energies of small organic molecules are predicted using kernel ridge regression. © 2015 Wiley Periodicals, Inc.

Models that combine quantum mechanics (QM) with machine learning (ML) aim to deliver the accuracy of QM at the speed of ML by interpolating between a feasible number of reference calculations. This hands-on tutorial introduces the reader to models based on kernel learning, an elegant, systematically nonlinear form of ML. Pseudocode, a reference implementation, and an example dataset are provided.

Quantum mechanics-based *ab initio* molecular dynamics (MD) simulation schemes offer an accurate and direct means to monitor the time evolution of materials. Nevertheless, the expensive and repetitive energy and force computations required in such simulations lead to significant bottlenecks. Here, we lay the foundations for an accelerated *ab initio* MD approach integrated with a machine learning framework. The proposed algorithm learns from previously visited configurations in a continuous and adaptive manner on-the-fly, and predicts (with chemical accuracy) the energies and atomic forces of a new configuration at a minuscule fraction of the time taken by conventional *ab initio* methods. Key elements of this new accelerated *ab initio* MD paradigm include representations of atomic configurations by numerical fingerprints, a learning algorithm to map the fingerprints to the properties, a decision engine that guides the choice of the prediction scheme, and requisite amount of *ab initio* data. The performance of each aspect of the proposed scheme is critically evaluated for Al in several different chemical environments. This work has enormous implications beyond *ab initio* MD acceleration. It can also lead to accelerated structure and property prediction schemes, and accurate force fields. © 2014 Wiley Periodicals, Inc.

The dynamical atomic-level evolution of typical chemical processes extends to timescales longer than nanoseconds—hard to access routinely using present *ab initio* methods. Here, an adaptive machine learning framework is proposed to significantly accelerate *ab initio* molecular dynamics simulations. The scheme learns to predict energies and atomic forces with unprecedented speed and accuracy from an initial *ab initio* dataset, and systematically expands its predictive capability on-the-fly by including newly encountered chemical environments in its training.

We introduce a fingerprint representation of molecules based on a Fourier series of atomic radial distribution functions. This fingerprint is unique (except for chirality), continuous, and differentiable with respect to atomic coordinates and nuclear charges. It is invariant with respect to translation, rotation, and nuclear permutation, and requires no preconceived knowledge about chemical bonding, topology, or electronic orbitals. As such, it meets many important criteria for a good molecular representation, suggesting its usefulness for machine learning models of molecular properties trained across chemical compound space. To assess the performance of this new descriptor, we have trained machine learning models of molecular enthalpies of atomization for training sets with up to 10 k organic molecules, drawn at random from a published set of 134 k organic molecules with an average atomization enthalpy of over 1770 kcal/mol. We validate the descriptor on all remaining molecules of the 134 k set. For a training set of 10 k molecules, the fingerprint descriptor achieves a mean absolute error of 8.0 kcal/mol. This is slightly worse than the performance attained using the Coulomb matrix, another popular alternative, reaching 6.2 kcal/mol for the same training and test sets. © 2015 Wiley Periodicals, Inc.

The accuracy of machine learning models of quantum mechanical observables of molecules hinges on the quality of the molecular representation. This article discusses necessary and desirable properties, as well as a Fourier series of atomic radial distribution functions, potentially useful as a unique molecular fingerprint. For heats of atomization of over 100,000 organic molecules, this fingerprint is shown to reach density functional theory level of accuracy.

We introduce and evaluate a set of feature vector representations of crystal structures for machine learning (ML) models of formation energies of solids. ML models of atomization energies of organic molecules have been successful using a Coulomb matrix representation of the molecule. We consider three ways to generalize such representations to periodic systems: (i) a matrix where each element is related to the Ewald sum of the electrostatic interaction between two different atoms in the unit cell repeated over the lattice; (ii) an extended Coulomb-like matrix that takes into account a number of neighboring unit cells; and (iii) an *ansatz* that mimics the periodicity and the basic features of the elements in the Ewald sum matrix using a sine function of the crystal coordinates of the atoms. The representations are compared for a Laplacian kernel with Manhattan norm, trained to reproduce formation energies using a dataset of 3938 crystal structures obtained from the Materials Project. For training sets consisting of 3000 crystals, the generalization error in predicting formation energies of new structures corresponds to (i) 0.49, (ii) 0.64, and (iii)
for the respective representations. © 2015 Wiley Periodicals, Inc.

Feature vector representations of crystal structures for machine learning models of formation energies of solids are evaluated. A representation previously found successful for molecules is generalized to periodic systems in three different ways. For training sets of 3000 crystal structures comprising all different elements, the best representation is estimated to predict formation energies of new materials with an average error of 0.37 eV/atom.

A method for nonlinear optimization with machine learning (ML) models, called nonlinear gradient denoising (NLGD), is developed, and applied with ML approximations to the kinetic energy density functional in an orbital-free density functional theory. Due to systematically inaccurate gradients of ML models, in particular when the data is very high-dimensional, the optimization must be constrained to the data manifold. We use nonlinear kernel principal component analysis (PCA) to locally reconstruct the manifold, enabling a projected gradient descent along it. A thorough analysis of the method is given via a simple model, designed to clarify the concepts presented. Additionally, NLGD is compared with the local PCA method used in previous work. Our method is shown to be superior in cases when the data manifold is highly nonlinear and high dimensional. Further applications of the method in both density functional theory and ML are discussed. © 2015 Wiley Periodicals, Inc.

When applying machine learning to model high-dimensional data, the gradient of the model can exhibit high levels of noise. In applications of machine learning to density functional theory to learn the kinetic energy as a functional of the density, the gradient is necessary to minimize the total energy and find a self-consistent ground-state density. A novel technique, called nonlinear gradient denoising, is developed to remove such noise.

Accurate approximations to density functionals have recently been obtained via machine learning (ML). By applying ML to a simple function of one variable without any random sampling, we extract the qualitative dependence of errors on hyperparameters. We find universal features of the behavior in extreme limits, including both very small and very large length scales, and the noise-free limit. We show how such features arise in ML models of density functionals. © 2015 Wiley Periodicals, Inc.

Machine learning with kernel ridge regression has recently been used to obtain approximations to density functionals. These approximations have been highly accurate for one-dimensional systems including the spinless fermion in a box and orbital-free bond breaking between diatomics. This article explores the behavior of machine learning models with respect to the properties of the kernel. Machine learning methods are tested on a simple function in order to elucidate this dependence.

Recent advances in quantum mechanical (QM)-based molecular dynamics (MD) simulations have used machine-learning (ML) to predict, rather than recalculate, QM-accurate forces in atomic configurations sufficiently similar to previously encountered ones. Here, we discuss how ML approaches can be deployed within large-scale QM/MM materials simulations on massively parallel supercomputers, making QM zones of atoms routinely attainable. We argue that the ML approach allows computational effort to be concentrated on the most chemically active subregions of the QM zone, significantly improving the overall efficiency of the simulation. We thus propose a novel method to partition large QM regions into multiple subregions, which can be computed in parallel to achieve optimal scaling. Then we review a recently proposed QM/ML MD scheme (Z. Li, J.R. Kermode, A. De Vita Phys. Rev. Lett., 2015, 114, 096405), discussing how this could be efficiently combined with QM-zone partitioning. © 2015 Wiley Periodicals, Inc.

A quantum mechanics/molecular mechanics (QM/MM) computational approach is necessary to describe many phenomena in nature. The scaling of this approach on petascale computers can be achieved through a new ensemble parallel scheme. Recent developments show that information efficient machine-learning methods can be used to predict QM-accurate forces. A framework for combining the two is proposed for maximally efficient QM/MM simulations.