Parallel implementation of divide-and-conquer semiempirical quantum chemistry calculations

We have implemented a parallel divide-and-conquer method for semiempirical quantum mechanical calculations. The standard message passing library, the message passing interface (MPI), was used. In this parallel version, the memory needed to store the Fock and density matrix elements is distributed among the processors. This memory distribution solves the problem of demanding requirement of memory for very large molecules. While the parallel calculation for construction of matrix elements is straightforward, the parallel calculation of Fock matrix diagonalization is achieved via the divide-and-conquer method. Geometry optimization is also implemented with parallel gradient calculations. The code has been tested on a Cray T3E parallel computer, and impressive speedup of calculations has been achieved. Our results indicate that the divide-and-conquer method is efficient for parallel implementation. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1101–1109, 1998     

Classical probability matrix: Prediction of quantum-state distributions by a moment analysis of classical trajectories

A new method is presented for extracting approximate quantum mechanical state-to-state transition probabilities from the results of classical trajectory calculations. The method recognizes quantum discreteness by dealing with the quantum mechanical probability matrix, but all dynamical quantities are evaluated by classical mechanics. It is illustrated by application to the linear atom-diatom collision (vibrational excitation); it is capable of treating both classically allowed and classically forbidden processes.     

Quantum crystallography applied to crystalline maleic anhydride

Quantum crystallography (QCr) is a term that concerns techniques for using crystallographic information to enhance quantum mechanical calculations and the information derived from them. In our approach to QCr, we use molecular orbitals and a single‐determinant density matrix formalism to develop a quantum mechanical model. Our initial application to a test material, crystalline maleic anhydride, involved the adjustment of the elements in the density (projector) matrix and some others in the quantum mechanical model. The purpose was to optimize the fit between the experimental structure factor magnitudes and the values of those magnitudes obtained from the quantum mechanical model. The adjustment of the projector matrix preserved the idempotency and normalization properties of the matrix. In this application, it was also found that it was necessary to correct the X‐ray diffraction data for systematic errors. An effective statistical method for doing this was developed from quantum mechanical theory. There were a number of special features of this investigation that emerged as it progressed. The mirror plane in maleic anhydride, for example, was quite useful because, in the absence of significant interactions between the molecules in the crystal, charge distributions on both sides of the mirror plane should be essentially the same. Deviations raised questions that resulted in improved procedures. The quality of theoretical results as a function of basis set and mode of calculation is also part of this investigation. One result of the information obtained from various aspects of this study is the potential for greater efficiency in the procedures and calculations. The calculations for maleic anhydride based on its structure concern the number of electrons per atom, various energies, and electron density contours. Related theoretical calculations based on geometry optimization were also made. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 439–450, 1999     

Vibrational matrix elements of the quadrupole moment functions of H2, N2 and CO

The quadrupole moment functions (molecular quadrupole moment versus internuclear distance) have been determined by quantum mechanical calculations for H₂ (by Kolos and Wolniewicz), N₂ (by Wahl and Nesbet), and CO (by Nesbet). These functions are used with numerical vibrational wave functions to compute matrix elements which are useful for calculations of scattering cross sections, energy transfer rates and excitation probabilities, and infrared intensities of forbidden bands.     

Finite group theory for large systems. 3. Symmetry-generation of reduced matrix elements for icosahedral C(20) and C(60) molecules

This paper uses symmetry-generation to simplify the determination of Hamiltonian reduced matrix elements. It is part of a series on using computers to apply finite group theory to quantum mechanical calculations on large systems. Symmetry-generation is an expression of the whole molecule as a sum of symmetry transformations on a smaller reference structure. Then on a suitably-conditioned symmetry-adapted basis, the reduced matrix elements of the Hamiltonian are averages of certain elements of the simpler reference structure matrix. The smaller the reference structure, the greater is the computational savings. Single atom reference structures are used here for the Hückel treatment of icosahedral C(20) and C(60) fullerenes. The analytical power of this approach is illustrated by determining the two bond lengths of C(60) from spectral data.     

On the importance of the classically forbidden region in calculations of the relaxation rate for high-frequency vibrations: a model calculation

The quantum mechanical relaxation rate for a high-frequency vibrational mode is evaluated for a one-dimensional model system having two diatomic molecules involved in a collinear collision. The thermally averaged rate is obtained as an integral over energies for the relative translation of the two molecules. These calculations show that energies several times K(B)T make the largest contributions to the rate. Several orders of magnitude of cancellation due to phase interference is found in the evaluation of the coupling matrix elements between the initial and final states, and this is one of the main factors leading to the very small value for the relaxation rate. The region near the classical turning point in the relative translational motion of the colliding molecules dominates the calculation of the contribution to the rate at each energy. Calculations using low-order expansions of the translational potential energy and the interstate coupling about this turning point provide good approximations to the exact quantum mechanical rate. This suggests a possible method for performing calculations of the rate by means of realistic simulations of liquid systems.     

Electronic coupling matrix elements from charge constrained density functional theory calculations using a plane wave basis set

We present a plane wave basis set implementation for the calculation of electronic coupling matrix elements of electron transfer reactions within the framework of constrained density functional theory (CDFT). Following the work of Wu and Van Voorhis [J. Chem. Phys. 125, 164105 (2006)], the diabatic wavefunctions are approximated by the Kohn-Sham determinants obtained from CDFT calculations, and the coupling matrix element calculated by an efficient integration scheme. Our results for intermolecular electron transfer in small systems agree very well with high-level ab initio calculations based on generalized Mulliken-Hush theory, and with previous local basis set CDFT calculations. The effect of thermal fluctuations on the coupling matrix element is demonstrated for intramolecular electron transfer in the tetrathiafulvalene-diquinone (Q-TTF-Q(-)) anion. Sampling the electronic coupling along density functional based molecular dynamics trajectories, we find that thermal fluctuations, in particular the slow bending motion of the molecule, can lead to changes in the instantaneous electron transfer rate by more than an order of magnitude. The thermal average, (<|H(ab)|(2)>)(1/2)=6.7 mH, is significantly higher than the value obtained for the minimum energy structure, |H(ab)|=3.8 mH. While CDFT in combination with generalized gradient approximation (GGA) functionals describes the intermolecular electron transfer in the studied systems well, exact exchange is required for Q-TTF-Q(-) in order to obtain coupling matrix elements in agreement with experiment (3.9 mH). The implementation presented opens up the possibility to compute electronic coupling matrix elements for extended systems where donor, acceptor, and the environment are treated at the quantum mechanical (QM) level.     

Conformer Pair Contributions to Optical Rotations in a Series of Chiral Linear Aliphatic Alcohols

The chain length effect of four chiral aliphatic alcohols,(S)-2-butanol,(S)-2-pentanol,(S)-2-hexanol and (S)-2-heptanol,on their specific optical rotations(OR)was studied experimentally and theoretically via quantum theory.Many conformations of each chiral alcohol exist as conformer pairs in solution.The OR sum from these pairs of conformers has much smaller contributions to OR values than that contributed by the most stable conformation. These four alcohols'OR values were also investigated using the matrix model,which employs each substituent's comprehensive mass,radii,electronegativity and symmetry number as the elements in the matrix.These are all particle properties.This matrix determinant is proportional to its OR values within a closely related structural series of chiral compounds.The experimental OR values and the matrix determinants of these four alcohols were compared with the predicted OR values obtained from quantum theory wave functions.The ORs predicted by the matrix method, which is based on particle function statistics,agreed with the results from quantum theory.The agreement between OR predictions by the matrix method and DFT calculations illustrates the wave-particle duality of polarized light that is operating in these predictions.     

First ultraviolet absorption band of methane: an ab initio study

Quantum mechanical calculations of the cross sections for photodissociation of CH4 and CD4 in the 1t2-->3s band are presented. The potential energy surfaces for the three states correlating with the 1 1T2 state at tetrahedral geometries are calculated. The elements of the (3x3) matrix representing the electronic Hamiltonian in the diabatic basis are expanded in powers of nuclear coordinates, up to the second order. The expansion coefficients are based on accurate multireference configuration interaction calculations. The electronically nonadiabatic dynamics is treated with the multiconfiguration time-dependent Hartree approach. All nine internal degrees of methane are included in the quantum dynamics simulations. The calculated cross section agrees well with experiment. Semiclassical calculations using the reflection principle suggest that the peaks in the spectrum correspond to the three adiabatic electronic states correlating with the 1 1T2 state at Td geometries. However, the non-Born-Oppenheimer terms in the Hamiltonian have a strong effect on the positions of the peaks in the absorption spectrum. The results of semiclassical calculations, which neglect these terms, are therefore quite different from the accurate quantum results and experiment.     

The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors

This work demonstrates that the quantum mechanical moments of a state described by the density matrix correspond to discrete spherical harmonic moments of the classical multipole expansion of the spatial distribution of the angular momentum vectors. For the diagonal density matrix elements, this work exploits the fact that the quantum mechanical vector coupling (Clebsch-Gordan) coefficients become increasingly accurate discrete representations of spherical harmonics as j increases. A Schwinger-type basis accounts for nonaxially symmetric angular distributions, which result in nonzero off-diagonal elements of the density matrix. The resulting discrete minimum uncertainty picture of the classical moments has a stringent equivalence with the quantum mechanical one for all j and provides an unambiguous connection for the classical and quantum moments in the large j limit. The equivalence is numerically tested for simple models, and there is a satisfying equivalence even for small j. Applications, implications, and extensions are indicated, and the relevance of this work for the interpretation of classical mechanical simulations of inelastic and reactive molecular collisions will be documented elsewhere.     

Exact quantum mechanical reaction probabilities for the collinear H + H2 reaction on a porter-karplus potential energy surface

Exact quantum mechanical calculation of the reaction probability for the collinear H + H₂ reaction on a Porter-Karplus potential energy surface are carried out by the finite-difference boundary value method at 6 energes in the threshold region and compared to close coupling, distorted wave, classical S matrix, transition state theory, and vibrational adiabatic calculations.     

Spin-orbit mechanism of predissociation in the Wulf band of ozone

Previously calculated resonance widths of the ground vibrational levels in the electronic states 1 (3)A" ((3)A(2)) and 1 (3)A' ((3)B(2)), which belong to the Wulf band system of ozone, are significantly smaller than observed experimentally. We demonstrate that predissociation is drastically enhanced by spin-orbit coupling between 1 (3)A"/X (1)A' and 1 (3)A'/1 (3)A". Multistate quantum mechanical calculations using ab initio spin-orbit coupling matrix elements give linewidths of optically bright components of the right order of magnitude.     

Derivation of closed analytical expressions for Rosen–Morse Franck–Condon factors

The expressions for Rosen–Morse Franck–Condon factors derived previously yield a double sum with alternating terms. For higher values of the quantum number the numerical calculation of the Franck–Condon factors by electronic computers using these expressions leads to numerical overflow inspite of the use of double-precision (32 digits) arithmetic. High values for the quantum number in the final ground state of the Rosen–Morse potential occur in molecular nonradiative rate calculations. Furthermore, the expressions show a lack of clearness with respect to the parameters of the potential. For out-of-plane modes exact closed form expressions and exact recurrence relations are derived. Asymptotic expressions for the matrix elements are calculated. Exact closed form expressions for matrix elements with quantum numbers which correspond to regions close to the dissociation barrier are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Reaction path potential for complex systems derived from combined ab initio quantum mechanical and molecular mechanical calculations

Combined ab initio quantum mechanical and molecular mechanical calculations have been widely used for modeling chemical reactions in complex systems such as enzymes, with most applications being based on the determination of a minimum energy path connecting the reactant through the transition state to the product in the enzyme environment. However, statistical mechanics sampling and reaction dynamics calculations with a combined ab initio quantum mechanical (QM) and molecular mechanical (MM) potential are still not feasible because of the computational costs associated mainly with the ab initio quantum mechanical calculations for the QM subsystem. To address this issue, a reaction path potential energy surface is developed here for statistical mechanics and dynamics simulation of chemical reactions in enzymes and other complex systems. The reaction path potential follows the ideas from the reaction path Hamiltonian of Miller, Handy and Adams for gas phase chemical reactions but is designed specifically for large systems that are described with combined ab initio quantum mechanical and molecular mechanical methods. The reaction path potential is an analytical energy expression of the combined quantum mechanical and molecular mechanical potential energy along the minimum energy path. An expansion around the minimum energy path is made in both the nuclear and the electronic degrees of freedom for the QM subsystem internal energy, while the energy of the subsystem described with MM remains unchanged from that in the combined quantum mechanical and molecular mechanical expression and the electrostatic interaction between the QM and MM subsystems is described as the interaction of the MM charges with the QM charges. The QM charges are polarizable in response to the changes in both the MM and the QM degrees of freedom through a new response kernel developed in the present work. The input data for constructing the reaction path potential are energies, vibrational frequencies, and electron density response properties of the QM subsystem along the minimum energy path, all of which can be obtained from the combined quantum mechanical and molecular mechanical calculations. Once constructed, it costs much less for its evaluation. Thus, the reaction path potential provides a potential energy surface for rigorous statistical mechanics and reaction dynamics calculations of complex systems. As an example, the method is applied to the statistical mechanical calculations for the potential of mean force of the chemical reaction in triosephosphate isomerase.     

A gauge-independent zeroth-order regular approximation to the exact relativistic Hamiltonian--formulation and applications

A simple modification of the zeroth-order regular approximation (ZORA) in relativistic theory is suggested to suppress its erroneous gauge dependence to a high level of approximation. The method, coined gauge-independent ZORA (ZORA-GI), can be easily installed in any existing nonrelativistic quantum chemical package by programming simple one-electron matrix elements for the quasirelativistic Hamiltonian. Results of benchmark calculations obtained with ZORA-GI at the Hartree-Fock (HF) and second-order Moller-Plesset perturbation theory (MP2) level for dihalogens X(2) (X=F,Cl,Br,I,At) are in good agreement with the results of four-component relativistic calculations (HF level) and experimental data (MP2 level). ZORA-GI calculations based on MP2 or coupled-cluster theory with single and double perturbations and a perturbative inclusion of triple excitations [CCSD(T)] lead to accurate atomization energies and molecular geometries for the tetroxides of group VIII elements. With ZORA-GI/CCSD(T), an improved estimate for the atomization energy of hassium (Z=108) tetroxide is obtained.     

NMR spectroscopy of intermetallic compounds: an experimental and theoretical approach to local atomic arrangements in binary gallides

The results of the investigation of MGa(2) with M = Ca, Sr, Ba and of MGa(4) with M = Na, Ca, Sr, Ba by a combined application of NMR spectroscopy and quantum mechanical calculations are comprehensively evaluated. The electric-field gradient (EFG) was identified as the most reliable measure to study intermetallic compounds, since it is accessible with high precision by quantum mechanical calculations and, for nuclear spin I>1/2, by NMR spectroscopy. The EFG values obtained by NMR spectroscopy and quantum mechanical calculations agree very well for both series of investigated compounds. A deconvolution of the calculated EFGs into their contributions reveals its sensitivity to the local environment of the atoms. The EFGs of the investigated di- and tetragallides are dominated by the population of the p(x)-, p(y)-, and p(z)-like states of the Ga atoms. A general combined approach for the investigation of disordered intermetallic compounds by application of diffraction methods, NMR spectroscopy, and quantum mechanical calculations is suggested. This scheme can also be applied to other classes of crystalline disordered inorganic materials.     

Generalized formulation of quantum and classical crystallography using Green's functions

Quantum crystallography is a methodology by which structural information about a crystalline material obtained from X‐ray crystallography is combined with quantum mechanical methods. The objective is to enhance the data obtained from the X‐ray diffraction experiment, which are related to the atomic structure of the crystal, and to predict the properties and efficacy of those chemical compounds from which the crystals are derived. One approach in quantum crystallography is to use a projector matrix with a normalized trace. In this approach, quantum mechanical parameters in the projector matrix are fit into crystallographic data. During this fitting, the properties of the projector matrix called idempotency and normalization are used. Throughout this implementation procedure, Clinton's iteration scheme has been used in addition to the least‐squares technique. The purpose of the present study is to generalize Clinton's iterative equations in quantum crystallography by means of single‐particle Green's functions with the aid of the equal atoms model in the theory of direct methods. Convergency characters of the novel iterative equations are discussed by the steepest descent procedure. Furthermore, whether the calculations are valid in nonorthogonal bases was also examined. The iteration schemes widely used in quantum crystallography have been generalized but, in addition, the generalized expressions relating to the phase determination procedure and the probabilities of the sign relations between the structure factors are obtained and discussed comprehensively. The phrase order of crystallography has been put forward as a new concept. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Mixed time slicing in path integral simulations

A simple and efficient scheme is presented for using different time slices for different degrees of freedom in path integral calculations. This method bridges the gap between full quantization and the standard mixed quantum-classical (MQC) scheme and, therefore, still provides quantum mechanical effects in the less-quantized variables. Underlying the algorithm is the notion that time slices (beads) may be "collapsed" in a manner that preserves quantization in the less quantum mechanical degrees of freedom. The method is shown to be analogous to multiple-time step integration techniques in classical molecular dynamics. The algorithm and its associated error are demonstrated on model systems containing coupled high- and low-frequency modes; results indicate that convergence of quantum mechanical observables can be achieved with disparate bead numbers in the different modes. Cost estimates indicate that this procedure, much like the MQC method, is most efficient for only a relatively few quantum mechanical degrees of freedom, such as proton transfer. In this regime, however, the cost of a fully quantum mechanical simulation is determined by the quantization of the least quantum mechanical degrees of freedom.