The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

Sum and density of states of polyatomic systems with hindered rotors

The density or sum of states for a collection of independent oscillators, free rotors, and one-dimensional hindered rotors is obtained with good accuracy by numerical inversion of the corresponding total partition function by the method of steepest descents. The hindered-rotor partition functions are used in both classical and quantum forms, the latter in the approximation proposed by Truhlar [J. Comput. Chem., 12, 266 (1991)]. The numerical inversion compares well with analytical results obtained in a simple artificial case and also with an exact count of states in a large ethane-like system. Inversion of the hindered-rotor classical partition function is shown to lead to a somewhat different energy dependence of the sum or density of states, relative to the quantum counterpart, which is considered to be a more realistic representation. The routines presented are simple and fast enough to be of use in microcanonical rate calculations. © 1996 by John Wiley & Sons, Inc.     

Tight-binding "dihedral orbitals" approach to the degree of folding of macromolecular chains

We develop a tight-binding molecular approach to quantify the degree of folding of a macromolecular chain. This approach is based on the linear combination of "dihedral" orbitals to give molecular orbitals (LCDO-MO). The dihedral orbitals are a set of orbitals situated in each dihedral angle of the chain. The LCDO-MO approach remains basically topological, and we display its direct relation to known graph theoretical concepts. Using this approach, we define the dihedral electronic energy and the dihedral electronic partition function of a linear macromolecular chain. We show that the partition function per dihedral angle quantifies the degree of folding of the dihedral graph. We analyze the empirical relationship between these two functions by using a series of 100 proteins. We also study the relation between these two functions and the percentages of secondary structure for these proteins. Finally, we illustrate the use of the dihedral energy and the partition function in structure-property studies of proteins by analyzing the binding of steroids to DB3 antibody.     

Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

Path integral approach to correlation energy

The Feynman path integral method is applied to the many-electron problem of quantum chemistry. We begin with investigating the partition function of the system in question; then, “a classical path of electron” that corresponds to the Hartree–Fock approximation is obtained by minimizing the thermodynamic potential of the system with respect to the electron coordinate. The next-order approximation is obtained by evaluating the deviation from this classical path, which is approximately written by an easily integrable Gaussian integral. The result is expected to be the random-phase approximation. As numerical examples, the hydrogen molecule and butadiene are treated. © 1994 John Wiley & Sons, Inc.     

Transport properties of normal liquid helium: comparison of various methodologies

We revisit the problem of self-diffusion in normal liquid helium above the lambda transition. Several different methods are applied to compute the velocity autocorrelation function. Since it is still impossible to determine the exact result for the velocity autocorrelation function from simulation, we appeal to the computation of short-time moments to determine the accuracy of the different approaches at short times. The main conclusion reached from our study is that both the quantum mode-coupling theory and the numerical analytic continuation approach must be regarded as a viable and competitive methods for the computation of dynamical properties of quantum systems.     

Analysis of Hydrogen Tunneling in an Enzyme Active Site using von Neumann Measurements

We build on our earlier quantum wavepacket study of hydrogen transfer in the biological enzyme, soybean lipoxygenase-1, by using von Neumann quantum measurement theory to gain qualitative insights into the transfer event. We treat the enzyme active site as a measurement device which acts on the tunneling hydrogen nucleus via the potential it exerts at each configuration. A series of changing active site geometries during the tunneling process effects a sequential projection of the initial, reactant state onto the final, product state. We study this process using several different kinds of von Neumann measurements and show how a discrete sequence of such measurements not only progressively increases the projection of the hydrogen nuclear wavepacket onto the product side but also favors proton over deuteron transfer. Several qualitative features of the hydrogen tunneling problem found in wavepacket dynamics studies are also recovered here. These include the shift in the "transition state" towards the reactant as a result of nuclear quantization, greater participation of excited states in the case of deuterium, and presence of critical points along the reaction coordinate that facilitate hydrogen and deuterium transfer and coincide with surface crossings. To further "tailor" the dynamics, we construct a perturbation to the sequence of measurements, that is a perturbation to the dynamical sequence of active site geometry evolution, which leads us to insight on the existence of sensitive regions of the reaction profile where subtle changes to the dynamics of the active site can have an effect on the hydrogen and deuterium transfer process.     

Generalized Gibbs state with modified Redfield solution: exact agreement up to second order

A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.     

Fractionation of peptide with disulfide bond for quantum mechanical calculation of interaction energy with molecules

We present a computational study of a recently developed molecular fractionation with conjugated caps (MFCC) method for application to peptide/protein that has disulfide bonds. Specifically, we employ the MFCC approach to generate peptide fragments in which a disulfide bond is cut and a pair of conjugated caps are inserted. The method is tested on two peptides interacting with a water molecule. The first is a dipeptide consisting of two cysteines (Cys-Cys) connected by a disulfide bond and the second is a seven amino acid peptide consisting of Gly-Cys-Gly-Gly-Gly-Cys-Gly with a disulfide cross link. One-dimensional peptide-water potential curves are computed using the MFCC method at various ab initio levels for a number of interaction geometries. The calculated interaction energies are found to be in excellent agreement with the results obtained from the corresponding full system ab initio calculations for both peptide/water systems. The current study provides further numerical support for the accuracy of the MFCC method in full quantum mechanical calculation of protein/peptide that contains disulfide bonds.     

Geometries and binding energies of small beryllium clusters using various levels of approximations for exchange and correlation: a comparative study

Electronic structure and geometries of small (n≦4) neutral, singly, and doubly ionized clusters of beryllium atoms have been studied using ab initio quantum chemical and local density techniques. In the quantum chemical method the exchange potential is treated exactly in the Hartree-Fock scheme while the correlation correction is incorporated through perturbative procedure. In the density functional approach the exchange and correlation potentials are treated in the local spin density approximation. To facilitate an unambiguous comparison between these methods we have used the same basis functions and numerical procedures. All the investigations yield nearly identical geometries. However, the binding energies using these methods can vary by as much as 2 eV and this variation is comparable to what one obtains using different basis functions.     

H/D isotope effect on porphine and porphycene molecules with multicomponent hybrid density functional theory

To analyze the H/D isotope effect on porphine and porphycene molecules including the protonic/deuteronic quantum nature and electron correlation efficiently, the authors have developed the new scheme of the multicomponent hybrid density functional theory [MC_(HF+DFT)]. The optimized geometries of porphine, porphycene, and these deuterated isotopomers by our MC_(HF+DFT) method are in good agreement with the experimental "high-symmetric" structures, contrary to the "low-symmetric" geometries optimized by pure multicomponent Hartree-Fock method. The optimized geometries for HD-porphine and HD-porphycene molecules, in which an inner hydrogen is replaced to a deuterium, are found to be low symmetric. Such drastic geometrical change induces the electronic polarization, and gives rise to the slight dipole moment values in these HD species. Their results clearly indicate that the difference of the nuclear quantum nature between inner proton and inner deuteron directly influences the molecular geometry and electronic structure.     

Lennard-Jones parameters for the combined QM/MM method using the B3LYP/6-31G*/AMBER potential

A combined DFT quantum mechanical and AMBER molecular mechanical potential (QM/MM) is presented for use in molecular modeling and molecular simulations of large biological systems. In our approach we evaluate Lennard-Jones parameters describing the interaction between the quantum mechanical (QM) part of a system, which is described at the B3LYP/6-31+G* level of theory, and the molecular mechanical (MM) part of the system, described by the AMBER force field. The Lennard-Jones parameters for this potential are obtained by calculating hydrogen bond energies and hydrogen bond geometries for a large set of bimolecular systems, in which one hydrogen bond monomer is described quantum mechanically and the other is treated molecular mechanically. We have investigated more than 100 different bimolecular systems, finding very good agreement between hydrogen bond energies and geometries obtained from the combined QM/MM calculations and results obtained at the QM level of theory, especially with respect to geometry. Therefore, based on the Lennard-Jones parameters obtained in our study, we anticipate that the B3LYP/6-31+G*/AMBER potential will be a precise tool to explore intermolecular interactions inside a protein environment.     

Multiple dot-in-rod PbS/CdS heterostructures with high photoluminescence quantum yield in the near-infrared

Pb cations in PbS quantum rods made from CdS quantum rods by successive complete cationic exchange reactions are partially re-exchanged for Cd cations. Using STEM-HAADF, we show that this leads to the formation of unique multiple dot-in-rod PbS/CdS heteronanostructures, with a photoluminescence quantum yield of 45-55%. We argue that the formation of multiple dot-in-rods is related to the initial polycrystallinity of the PbS quantum rods, where each PbS crystallite transforms in a separate PbS/CdS dot-in-dot. Effective mass modeling indicates that electronic coupling between the different PbS conduction band states is feasible for the multiple dot-in-rod geometries obtained, while the hole states remain largely uncoupled.     

Semiclassical glory analyses in the time domain for the H + D2(v(i) = 0, j(i) = 0) → HD(v(f) = 3, j(f) = 0) + D reaction

We make the first application of semiclassical (SC) techniques to the plane-wavepacket formulation of time-domain (T-domain) scattering. The angular scattering of the state-to-state reaction, H + D(2)(v(i) = 0, j(i) = 0) → HD(v(f) = 3, j(f) = 0) + D, is analysed, where v and j are vibrational and rotational quantum numbers, respectively. It is proved that the forward-angle scattering in the T-domain, which arises from a delayed mechanism, is an example of a glory. The SC techniques used in the T-domain are: An integral transitional approximation, a semiclassical transitional approximation, a uniform semiclassical approximation (USA), a primitive semiclassical approximation and a classical semiclassical approximation. Nearside-farside (NF) scattering theory is also employed, both partial wave and SC, since a NF analysis provides valuable insights into oscillatory structures present in the full scattering pattern. In addition, we incorporate techniques into the SC theory called "one linear fit" and "two linear fits", which allow the derivative of the quantum deflection function, Θ?(')(J), to be estimated when Θ?J exhibits undulations as a function of J, the total angular momentum variable. The input to our SC analyses is numerical scattering (S) matrix data, calculated from accurate quantum collisional calculations for the Boothroyd-Keogh-Martin-Peterson potential energy surface No. 2, in the energy domain (E-domain), from which accurate S matrix elements in the T-domain are generated. In the E-domain, we introduce a new technique, called "T-to-E domain SC analysis." It half-Fourier transforms the E-domain accurate quantum scattering amplitude to the T-domain, where we carry out a SC analysis; this is followed by an inverse half-Fourier transform of the T-domain SC scattering amplitude back to the E-domain. We demonstrate that T-to-E USA differential cross sections (DCSs) agree well with exact quantum DCSs at forward angles, for energies where a direct USA analysis in the E-domain fails.     

Are AM1 ligand‐protein binding enthalpies good enough for use in the rational design of new drugs?

We have examined the performance of semiempirical quantum mechanical methods in solving the problem of accurately predicting protein-ligand binding energies and geometries. Firstly, AM1 and PM3 geometries and binding enthalpies between small molecules that simulate typical ligand-protein interactions were compared with high level quantum mechanical techniques that include electronic correlation (e.g., MP2 or B3LYP). Species studied include alkanes, aromatic systems, molecules including groups with hypervalent sulfur or with donor or acceptor hydrogen bonding capability, as well as ammonium or carboxylate ions. B3LYP/6-311+G(2d,p) binding energies correlated very well with the BSSE corrected MP2/6-31G(d) values. AM1 binding enthalpies also showed good correlation with MP2 values, and their systematic deviation is acceptable when enthalpies are used for the comparison of interaction energies between ligands and a target. PM3 otherwise gave erratic energy differences in comparison to the B3LYP or MP2 approaches. As one would expect, the geometries of the binding complexes showed the known limitations of the semiempirical and DFT methods. AM1 calculations were subsequently applied to a test set consisting of "real" protein active site-ligand complexes. Preliminary results indicate that AM1 could be a valuable tool for the design of new drugs using proteins as templates. This approach also has a reasonable computational cost. The ligand-protein X-ray structures were reasonably reproduced by AM1 calculations and the corresponding AM1 binding enthalpies are in agreement with the results from the "small molecules" test set.     

Path integral Monte Carlo method for ab initio calculation

The Feynman path integral method is applied to the many-electron problem. We first give new closure relations in terms of ordinary complex and real numbers, which could be derived from an arbitrary complete set of state vectors. Then, in the path integral form, the partition function of the system and the ensemble average of energy are explicitly expressed in terms of these closure relations. It is impossible to evaluate the path integral by direct numerical integrations because of its huge amount of integration variables. Therefore, we develop an algorithm by the Monte Carlo method with constraints corresponding to the normalization condition of states to calculate the required integral. Finally, the ensemble average of energy for the hydrogen molecule is explicitly evaluated by the quantum Monte Carlo method and results are compared with the result obtained by the ordinary full configuration interaction (CI) method. © 1996 John Wiley & Sons, Inc.     

Quantum statistical mechanics of closed‐ring molecular chains

A model for a closed‐ring unhindered three‐dimensional macromolecular chain, based on Quantum Mechanics, is presented. Upon starting from an exact non‐relativistic Hamiltonian operator, we integrate out all electronic degrees of freedom, in the Born‐Oppenheimer framework, giving rise to an effective vibro‐rotational Hamiltonian for the chain. Then, assuming a harmonic oscillator‐like vibrational potential between nearest‐neighbour atoms, the integration of the atomic radial degrees of freedom is carried in the limit of high frequencies. Thus, all bond lengths become fixed, including the one which makes the chain to become a closed ring. This formulation leads to a specific Hamiltonian for the remaining angular variables of the closed‐ring chain, and constitutes an alternative in comparison with standard Gaussian models, which do not. Use is made of a variational inequality by Peierls to find an approximate quantum partition function for the angular variables of the system. We then proceed to obtain approximately another representation for the angular partition function in the classical limit. Several features of the classical partition function are discussed.     

Non-equilibrium spin-boson model: counting statistics and the heat exchange fluctuation theorem

We focus on the non-equilibrium two-bath spin-boson model, a toy model for examining quantum thermal transport in many-body open systems. Describing the dynamics within the noninteracting-blip approximation equations, applicable, e.g., in the strong system-bath coupling limit and/or at high temperatures, we derive expressions for the cumulant generating function in both the Markovian and non-Markovian limits by energy-resolving the quantum master equation of the subsystem. For a Markovian bath, we readily demonstrate the validity of a steady-state heat exchange fluctuation theorem. In the non-Markovian limit a "weaker" symmetry relation generally holds, a general outcome of microreversibility. We discuss the reduction of this symmetry relation to the universal steady-state fluctuation theorem. Using the cumulant generating function, an analytic expression for the heat current is obtained. Our results establish the validity of the steady-state heat exchange fluctuation theorem in quantum systems with strong system-bath interactions. From the practical point of view, this study provides tools for exploring transport characteristics of the two-bath spin-boson model, a prototype for a nonlinear thermal conductor.