Several routes to the glassy states in the one component soft core system: revisited by molecular dynamics

Molecular dynamics simulations have been performed to study the glass transition for the soft core system with a pair potential φ(n)(r) = ε(σ∕r)(n) of n = 12. Using the compressibility factor, PV/Nk(B)T=P?(ρ*), its phase diagram can be represented as a function of a reduced density, ρ? = ρ(ε∕k(B)T)(3∕n), where ρ = Nσ(3)∕V. In the present work, NVE relaxations to the glassy or crystalline states starting from the unstable states in the phase diagram have been revisited in details and compared with other processes. Relaxation processes can be characterized by the time dependence of the dynamical compressibility factor (PV/Nk(B)T)(t)?(≡g(ρ(t)*)) on the phase diagram. In some cases, g(ρ(t)*) reached a crystal branch in the phase diagram; however, metastable states are found in many cases. With connecting points for the metastable states in the phase diagram, we can define a glass branch where the dynamics of particles are almost frozen. The structures observed there have common properties characterized as glasses. Although overlaps of glass forming process and nanocrystallization process are observed in some cases, these behaviors are distinguishable to each other by the characteristics of structures. There are several routes to the glass branch and we suggest that all of them are the glass transition.     

Molecular density of states from estimated vapor phase heat capacities

Heat capacity data between 298 and 1500K are used to derive a reduced set of apparent vibrational frequencies that can be used for estimation of molecular density of states, ρ(E). Estimates for a number of molecule and radical species, using a reduced set of three frequencies with noninteger degeneracies, are shown to compare favorably to direct count methods, which require specification of the complete frequency set. Use of the reduced set of three frequencies leads to significant improvement in calculations of ρ(E)/Q as compared to similar calculations which use only a single geometric- or arithmetic-mean frequency approximation. Since vapor phase heat capacity data of molecules and radicals can be estimated accurately by a group additivity formalism, this approach provides a method to estimate ρ(E) for use in calculations of pressure effects in unimolecular and chemical activation reaction systems. The accuracy of the ρ(E)/Q distributions obtained from heat capacity data makes this a viable method for those cases where the complete frequency distribution is not known. It is especially valuable for those cases where contributions to ρ(E) from internal rotors or low frequency vibrations such as inversions are not well known. This approach is useful for quantum RRK or inverse Laplace transform calculations of k(E) since no assignment of transition state properties is necessary. The reduced frequency set can also be combined with ΔH_f(298) and S(298) to provide a compact data set to describe thermodynamic properties at any temperature. © 1997 John Wiley & Sons, Inc.     

Laser control of molecular excitations in stochastic dissipative media

In the present work, ideas for controlling photochemical reactions in dissipative environments using shaped laser pulses are presented. New time-local control algorithms for the stochastic Schro?dinger equation are introduced and compared to their reduced density matrix analog. The numerical schemes rely on time-dependent targets for guiding the reaction along a preferred path. The methods are tested on the vibrational control of adsorbates at metallic surfaces and on the ultrafast electron dynamics in a strong dissipative medium. The selective excitation of the specific states is achieved with improved yield when using the new algorithms. Both methods exhibit similar convergence behavior and results compare well with those obtained using local optimal control for the reduced density matrix. The favorable scaling of the methods allows to tackle larger systems and to control photochemical reactions in dissipative media of molecules with many more degrees of freedom.     

Hamiltonian matrix and reduced density matrix construction with nonlinear wave functions

An efficient procedure to compute Hamiltonian matrix elements and reduced one- and two-particle density matrices for electronic wave functions using a new graphical-based nonlinear expansion form is presented. This method is based on spin eigenfunctions using the graphical unitary group approach (GUGA), and the wave function is expanded in a basis of product functions (each of which is equivalent to some linear combination of all of the configuration state functions), allowing application to closed- and open-shell systems and to ground and excited electronic states. In general, the effort required to construct an individual Hamiltonian matrix element between two product basis functions H(MN) = M|H|N scales as theta (beta n4) for a wave function expanded in n molecular orbitals. The prefactor beta itself scales between N0 and N2, for N electrons, depending on the complexity of the underlying Shavitt graph. Timings with our initial implementation of this method are very promising. Wave function expansions that are orders of magnitude larger than can be treated with traditional CI methods require only modest effort with our new method.     

Linear scaling calculation of band edge states and doped semiconductors

Linear scaling methods provide total energy, but no energy levels and canonical wave functions. From the density matrix computed through the density matrix purification methods, we propose an order-N [O(N)] method for calculating both the energies and wave functions of band edge states, which are important for optical properties and chemical reactions. In addition, we also develop an O(N) algorithm to deal with doped semiconductors based on the O(N) method for band edge states calculation. We illustrate the O(N) behavior of the new method by applying it to boron nitride (BN) nanotubes and BN nanotubes with an adsorbed hydrogen atom. The band gap of various BN nanotubes are investigated systematically and the acceptor levels of BN nanotubes with an isolated adsorbed H atom are computed. Our methods are simple, robust, and especially suited for the application in self-consistent field electronic structure theory.     

Improved configuration space sampling: Langevin dynamics with alternative mobility

We present a new and efficient method for determining optimal configurations of a large number (N) of interacting particles. We use a coarse-grained stochastic Langevin equation in the overdamped limit to describe the dynamics of this system and replace the standard mobility by an effective space dependent inverse Hessian correlation matrix. Due to the analogy of the drift term in the Langevin equation and the update scheme in Newton's method, we expect accelerated dynamics or improved convergence in the convex part of the potential energy surface Phi. The stochastic noise term, however, is not only essential for proper thermodynamic sampling but also allows the system to access transition states in the concave parts of Phi. We employ a Broyden-Fletcher-Goldfarb-Shannon method for updating the local mobility matrix. Quantitative analysis for one and two dimensional systems shows that the new method is indeed more efficient than standard methods with constant effective friction. Due to the construction, our effective mobility adapts high values/low friction in configurations which are less optimal and low values/high friction in configurations that are more optimal.     

Efficient first-principles electronic dynamics

An efficient first-principles electronic dynamics method is introduced in this article. The approach we put forth relies on incrementally constructing a time-dependent Fock∕Kohn-Sham matrix using active space density screening method that reduces the cost of computing two-electron repulsion integrals. An adaptive stepsize control algorithm is developed to optimize the efficiency of the electronic dynamics while maintaining good energy conservation. A selected set of model dipolar push-pull chromophore molecules are tested and compared with the conventional method of direct formation of the Fock∕Kohn-Sham matrix. While both methods considered herein take on identical dynamical simulation pathways for the molecules tested, the active space density screening algorithm becomes much more computationally efficient. The adaptive stepsize control algorithm, when used in conjunction with the dynamically active space method, yields a factor of ～3 speed-up in computational cost as observed in electronic dynamics using the time dependent density functional theory. The total computational cost scales nearly linear with increasing size of the molecular system.     

An Anderson impurity model for efficient sampling of adiabatic potential energy surfaces of transition metal complexes

We present a model intended for rapid sampling of ground and excited state potential energy surfaces for first-row transition metal active sites. The method is computationally inexpensive and is suited for dynamics simulations where (1) adiabatic states are required "on-the-fly" and (2) the primary source of the electronic coupling between the diabatic states is the perturbative spin-orbit interaction among the 3d electrons. The model Hamiltonian we develop is a variant of the Anderson impurity model and achieves efficiency through a physically motivated basis set reduction based on the large value of the d-d Coulomb interaction U(d) and a Lanczos matrix diagonalization routine to solve for eigenvalues. The model parameters are constrained by fits to the partial density of states obtained from ab initio density functional theory calculations. For a particular application of our model we focus on electron transfer occurring between cobalt ions solvated by ammonium, incorporating configuration interaction between multiplet states for both metal ions. We demonstrate the capability of the method to efficiently calculate adiabatic potential energy surfaces and the electronic coupling factor we have calculated compares well to previous calculations and experiment. (     

Using gamma distributions to predict self-diffusivities and density of states of fluids confined in carbon nanotubes

The density of states of rare gas atoms confined in carbon nanotubes is analyzed using a recently proposed model based on gamma distributions [Krishnan and Ayappa, J. Chem. Phys., 124 144503 (2006)]. The inputs into the model are the 2nd and 4th frequency moments that are obtained from molecular dynamics simulations. The predicted density of states, velocity autocorrelation functions and self-diffusivities are compared with those obtained from molecular dynamics simulations, for different nanotube loadings and temperatures. All results are reported for argon confined in a (16,16) carbon nanotube. The model predictions are extremely accurate at intermediate reduced densities of rhosigma(3) = 0.3, 0.4, where the majority of the self-diffusivity predictions lie within 10% of the simulation results. Since the frequency moments can be also obtained from Monte Carlo simulations, the study suggests an alternate route to the system dynamics of strongly confined fluids.     

Density matrix equation for crystals

An idempotent density matrix in which the orbitals are expanded in Bloch sums is used in the density matrix equation for ρ₁(1,1′) to obtain an equation appropriate for a crystal. The general equation is presented as well as its simplification for the single-cell approximation and the nearest-neighbor approximation.     

Linear-scaling quantum mechanical methods for excited states

The poor scaling of many existing quantum mechanical methods with respect to the system size hinders their applications to large systems. In this tutorial review, we focus on latest research on linear-scaling or O(N) quantum mechanical methods for excited states. Based on the locality of quantum mechanical systems, O(N) quantum mechanical methods for excited states are comprised of two categories, the time-domain and frequency-domain methods. The former solves the dynamics of the electronic systems in real time while the latter involves direct evaluation of electronic response in the frequency-domain. The localized density matrix (LDM) method is the first and most mature linear-scaling quantum mechanical method for excited states. It has been implemented in time- and frequency-domains. The O(N) time-domain methods also include the approach that solves the time-dependent Kohn-Sham (TDKS) equation using the non-orthogonal localized molecular orbitals (NOLMOs). Besides the frequency-domain LDM method, other O(N) frequency-domain methods have been proposed and implemented at the first-principles level. Except one-dimensional or quasi-one-dimensional systems, the O(N) frequency-domain methods are often not applicable to resonant responses because of the convergence problem. For linear response, the most efficient O(N) first-principles method is found to be the LDM method with Chebyshev expansion for time integration. For off-resonant response (including nonlinear properties) at a specific frequency, the frequency-domain methods with iterative solvers are quite efficient and thus practical. For nonlinear response, both on-resonance and off-resonance, the time-domain methods can be used, however, as the time-domain first-principles methods are quite expensive, time-domain O(N) semi-empirical methods are often the practical choice. Compared to the O(N) frequency-domain methods, the O(N) time-domain methods for excited states are much more mature and numerically stable, and have been applied widely to investigate the dynamics of complex molecular systems.     

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.     

Quantum wavepacket ab initio molecular dynamics: generalizations using an extended Lagrangian treatment of diabatic states coupled through multireference electronic structure

We present a generalization to our previously developed quantum wavepacket ab initio molecular dynamics (QWAIMD) method by using multiple diabatic electronic reduced single particle density matrices, propagated within an extended Lagrangian paradigm. The Slater determinantal wavefunctions associated with the density matrices utilized may be orthogonal or nonorthogonal with respect to each other. This generalization directly results from an analysis of the variance in electronic structure with quantum nuclear degrees of freedom. The diabatic electronic states are treated here as classical parametric variables and propagated simultaneously along with the quantum wavepacket and classical nuclei. Each electronic density matrix is constrained to be N-representable. Consequently two sets of new methods are derived: extended Lagrangian-QWAIMD (xLag-QWAIMD) and diabatic extended Lagrangian-QWAIMD (DxLag-QWAIMD). In both cases, the instantaneous potential energy surface for the quantum nuclear degrees of freedom is constructed from the diabatic states using an on-the-fly nonorthogonal multireference formalism. By introducing generalized grid-based electronic basis functions, we eliminate the basis set dependence on the quantum nucleus. Subsequent reuse of the two-electron integrals during the on-the-fly potential energy surface computation stage yields a substantial reduction in computational costs. Specifically, both xLag-QWAIMD and DxLag-QWAIMD turn out to be about two orders of magnitude faster than our previously developed time-dependent deterministic sampling implementation of QWAIMD. Energy conservation properties, accuracy of the associated potential surfaces, and vibrational properties are analyzed for a family of hydrogen bonded systems.     

Inequalities for fermion density matrices

A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single-particle states is defined as the (reduced) k-state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one- and two-particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density-matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off-diagonal elements of the density matrices.     

Evaluation of the field-adapted ADMA approach: absolute and relative energies of crambin and derivatives

A large number of conformations and chemically modified variants of the protein crambin were used to extensively test the field-adapted adjustable density matrix assembler (FA-ADMA) method developed for ab initio quality quantum chemistry computations of proteins and other macromolecules, introduced in an earlier publication. In this method, the fuzzy density matrix fragmentation scheme of the original adjustable density matrix assembler (ADMA) method has been made more efficient by combining it with an approach of using point charges to approximate the effects of additional, distant parts of a given macromolecule in the quantum chemical calculation of each fragment. In this way, smaller parent molecules can be used for fragment generation, while achieving accuracy that can be obtained only with large parent molecules in the original ADMA method. Whereas in both methods the error relative to the Hartree-Fock result can be reduced below any threshold by choosing large enough parent molecules, this can be done more efficiently with the new method. In order to obtain reliable test results for the accuracy obtainable by the new method when compared to conventional Hartree-Fock calculations, we performed a large number of energy calculations for the protein crambin using various conformations available in the Protein Data Bank, various protonation states, and side chain mutations. Additionally, in order to test the performance of the method for protein-solvent interaction studies, the energy changes due to the formation of complexes with ethanol and single and multiple water molecules were investigated.     

Thermodynamic scaling of α-relaxation time and viscosity stems from the Johari-Goldstein β-relaxation or the primitive relaxation of the coupling model

By now it is well established that the structural α-relaxation time, τ(α), of non-associated small molecular and polymeric glass-formers obey thermodynamic scaling. In other words, τ(α) is a function Φ of the product variable, ρ(γ)/T, where ρ is the density and T the temperature. The constant γ as well as the function, τ(α) = Φ(ρ(γ)/T), is material dependent. Actually this dependence of τ(α) on ρ(γ)/T originates from the dependence on the same product variable of the Johari-Goldstein β-relaxation time, τ(β), or the primitive relaxation time, τ(0), of the coupling model. To support this assertion, we give evidences from various sources itemized as follows. (1) The invariance of the relation between τ(α) and τ(β) or τ(0) to widely different combinations of pressure and temperature. (2) Experimental dielectric and viscosity data of glass-forming van der Waals liquids and polymer. (3) Molecular dynamics simulations of binary Lennard-Jones (LJ) models, the Lewis-Wahnstr?m model of ortho-terphenyl, 1,4 polybutadiene, a room temperature ionic liquid, 1-ethyl-3-methylimidazolium nitrate, and a molten salt 2Ca(NO(3))(2)·3KNO(3) (CKN). (4) Both diffusivity and structural relaxation time, as well as the breakdown of Stokes-Einstein relation in CKN obey thermodynamic scaling by ρ(γ)/T with the same γ. (5) In polymers, the chain normal mode relaxation time, τ(N), is another function of ρ(γ)/T with the same γ as segmental relaxation time τ(α). (6) While the data of τ(α) from simulations for the full LJ binary mixture obey very well the thermodynamic scaling, it is strongly violated when the LJ interaction potential is truncated beyond typical inter-particle distance, although in both cases the repulsive pair potentials coincide for some distances.     

A master equation approach to the dynamics of zero electron kinetic energy (ZEKE) states and ZEKE spectroscopy

We have theoretically studied important dynamic processes involved in zero electron kinetic energy (ZEKE) spectroscopy using the density matrix method with the inverse Born-Oppenheimer approximation basis sets. In ZEKE spectroscopy, the ZEKE Rydberg states are populated by laser excitation (either a one- or two-photon process), which is followed by autoionizations and l-mixing due to a stray field. The discrimination field is then applied to ionize loosely bound electrons in the ZEKE states. This is followed by using the extraction field to extract electrons from the ZEKE levels which have a strength comparable to that of the extraction field. These extracted electrons are measured for the relative intensities of the ion states under investigation. The spectral positions are determined by the applied laser wavelength and modified by the extraction electric field. In this paper, all of these processes are conducted within the context of the density matrix method. The density matrix method can provide not only the dynamics of system's population and coherence (or phase) but also the rate constants of the processes involved in the ZEKE spectroscopy. Numerical examples are given to demonstrate the theoretical treatments.     

An adaptive pseudospectral method for wave packet dynamics

We solve the time-dependent Schro?dinger equation for molecular dynamics using a pseudospectral method with global, exponentially decaying, Hagedorn basis functions. The approximation properties of the Hagedorn basis depend strongly on the scaling of the spatial coordinates. Using results from control theory we develop a time-dependent scaling which adaptively matches the basis to the wave packet. The method requires no knowledge of the Hessian of the potential. The viability of the method is demonstrated on a model for the photodissociation of IBr, using a Fourier basis in the bound state and Hagedorn bases in the dissociative states. Using the new approach to adapting the basis we are able to solve the problem with less than half the number of basis functions otherwise necessary. We also present calculations on a two-dimensional model of CO(2) where the new method considerably reduces the required number of basis functions compared to the Fourier pseudospectral method.     

The density of states for interacting systems with inelastic transitions and its relation to the collision lifetime

The quantum mechanical excess density of states ρ(E) is shown to be given in terms of the S matrix by (i/2π) Tr{SdS⁺/dE}, giving the relationship ρ(E) = h⁻¹ Tr{Q}, where Q is the collision lifetime matrix of Smith.