Application of variational reduced-density-matrix theory to organic molecules

Variational calculation of the two-electron reduced-density matrix (2-RDM), using a new first-order algorithm [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)], is applied to medium-sized organic molecules. The calculations reveal systematic trends in the accuracy of the energy with well-known chemical concepts such as hybridization, electronegativity, and atomic size. Furthermore, correlation energies from hydrocarbon chains indicate that the calculation of the 2-RDM subject to two-positivity conditions is size extensive, that is, the energy grows linearly with the number of electrons. Because organic molecules have a well-defined set of functional groups, we employ the trends in energy accuracy of the functional groups to design a correction to the 2-RDM energy for an arbitrary organic molecule. We apply the 2-RDM calculations with the functional-group correction to a large set of organic molecules with different functional groups. Energies with millihartree accuracy are obtained both at equilibrium and nonequilibrium geometries.     

Simple Hamiltonians which exhibit drastic failures by variational determination of the two-particle reduced density matrix with some well known N-representability conditions

Calculations on small molecular systems indicate that the variational approach employing the two-particle reduced density matrix (2-RDM) as the basic unknown and applying the P, Q, G, T1, and T2 representability conditions provides an accuracy that is competitive with the best standard ab initio methods of quantum chemistry. However, in this paper we consider a simple class of Hamiltonians for which an exact ground state wave function can be written as a single Slater determinant and yet the same 2-RDM approach gives a drastically nonrepresentable result. This shows the need for stronger representability conditions than the mentioned ones.     

Molecular properties from perturbation theory: A Unified treatment of energy derivatives

The definition of a molecular property as a derivative of the electronic energy with respect to one or more applied perturbations is reviewed. The explicit enumeration of terms entering the derivative formulas is performed by considering in turn the various parameter spaces on which the energy and wave function depend. After deriving general expressions for first, second, and third derivatives for different types of perturbation, the parameter spaces involved in MCSCF and CI cases are identified and used to obtain expressions for the first and second derivatives. An example of an MCSCF third derivative is also given. In addition, the various equation systems defining the perturbed wave functions in each order are derived. Some attention is given to the efficient computer implementation of derivative calculations, and the present work is compared with that of other authors.     

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong.     

New N-representability results with symmetry in molecular quantum chemistry

We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $\rho \Psi $ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Görling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.     

Molecular dynamics with stochastic boundary conditions

We present and illustrate a simple approach for carrying out molecular dynamics simulations subject to stochastic boundary conditions. Methods of this type are expected to be useful in the study of chemical reactions and other localized processes in dense media.     

Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy

It is demonstrated that frequency-dependent response functions can conveniently be derived from the time-averaged quasienergy. The variational criteria for the quasienergy determines the time-evolution of the wave-function parameters and the time-averaged time-dependent Hellmann–Feynman theorem allows an identification of response functions as derivatives of the quasienergy. The quasienergy therefore plays the same role as the usual energy in time-independent theory, and the same techniques can be used to obtain computationally tractable expressions for response properties, as for energy derivatives in time-independent theory. This includes the use of the variational Lagrangian technique for obtaining expressions for molecular properties in accord with the 2n+1 and 2n+2 rules. The derivation of frequency-dependent response properties becomes a simple extension of variational perturbation theory to a Fourier component variational perturbation theory. The generality and simplicity of this approach are illustrated by derivation of linear and higher-order response functions for both exact and approximate wave functions and for both variational and nonvariational wave functions. Examples of approximate models discussed in this article are coupled-cluster, self-consistent field, and second-order Møller–Plesset perturbation theory. A discussion of symmetry properties of the response functions and their relation to molecular properties is also given, with special attention to the calculation of transition- and excited-state properties. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 1–52, 1998     

Molecular properties from coupled-cluster Brueckner orbitals

We investigate to which extent a single determinant made up from orbitals obtained by a Brueckner coupled-cluster doubles calculation is able to reproduce correlated one-electron properties. It is shown that dipole and quadrupole moments and radial expectation values compare quite well with BCCD finite-field results for a test selection of nine molecules enclosing HF, H₂O, NH₃, CO, N₂, NO⁺, HCN, CuH and CH₃OH and three rare-gas atoms He, Ne and Ar. Furthermore, we find that even second-order properties such as dipole and quadrupole polarizabilities are reproduced fairly well when determined as first derivatives of the corresponding Brueckner orbital expectation values.     

Molecular binding energies from partition density functional theory

Approximate molecular calculations via standard Kohn-Sham density functional theory are exactly reproduced by performing self-consistent calculations on isolated fragments via partition density functional theory [P. Elliott, K. Burke, M. H. Cohen, and A. Wasserman, Phys. Rev. A 82, 024501 (2010)]. We illustrate this with the binding curves of small diatomic molecules. We find that partition energies are in all cases qualitatively similar and numerically close to actual binding energies. We discuss qualitative features of the associated partition potentials.     

Calculating rate constants with updated Hessians using variational transition state theory with multidimensional tunneling

Variational transition state theory with multidimensional tunneling (VTST/MT) has been used for calculating the rate constants of reactions. The updated Hessians have been used to reduce the computational costs for both geometry optimization and trajectory following procedures. In this paper, updated Hessians are used to reduce the computational costs while calculating the rate constants applying VTST/MT. Although we found that directly applying the updated Hessians will not generate good vibrational frequencies along the minimum energy path (MEP), however, we can either re-compute the full Hessian matrices at fixed intervals or calculate the Block Hessians, which is constructed by numerical one-side difference for the Hessian elements in the "critical" region and Bofill updating scheme for the rest of the Hessian elements. Due to the numerical instability of the Bofill update method near the saddle point region, we have suggested a simple strategy in which we follow the MEP until certain percentage of the classical barrier height from the barrier top with full Hessians computed and then performing rate constant calculation with the extended MEP using Block Hessians. This strategy results a mean unsigned percentage deviation (MUPD) around 10% with full Hessians computed till the point with 80% classical barrier height for four studied reactions. This proposed strategy is attractive not only it can be implemented as an automatic procedure but also speeds up the VTST/MT calculation via embarrassingly parallelization to a personal computer cluster.     

Molecular dynamics with helical periodic boundary conditions

Helical symmetry is often encountered in nature and thus also in molecular dynamics (MD) simulations. In many cases, an approximation based on infinite helical periodicity can save a significant amount of computer time. However, standard simulations with the usual periodic boundary conditions (PBC) are not easily compatible with it. In the present study, we propose and investigate an algorithm comprising infinitely propagated helicity, which is compatible with commonly used MD software. The helical twist is introduced as a parametric geometry constraint, and the translational PBC are modified to allow for the helical symmetry via a transitional solvent volume. The algorithm including a parallel code was implemented within the Tinker software. The viability of the helical periodic boundary conditions (HPBC) was verified in test simulations including α‐helical and polyproline II like peptide structures. For an insulin‐based model, the HPBC dynamics made it possible to simulate a fibrillar structure, otherwise not stable within PBC. © 2014 Wiley Periodicals, Inc.     

Equation of state for expanded fluid mercury: variational theory with many-body interaction

A variational associating fluid theory is proposed to describe equations of state for expanded fluid mercury. The theory is based on the soft-sphere variational theory, incorporating an ab initio diatomic potential and an attractive many-body potential; the latter is evaluated with quantum chemical methods and expressed as a function of the local atomic coordination number and the nearest-neighbor distance. The resultant equation of state can reproduce the observed gas-liquid coexistence curve with good accuracy, without introducing phenomenological effective pair potentials. Various thermodynamic quantities such as pressure, isocloric thermal pressure coefficient, adiabatic sound velocity, and specific heat are calculated over a wide density-temperature range and compared with available experimental data.     

Molecular theory of the viscoelāstic properties of concentrated polymer solutions

The viscoelastic properties of concentrated polymer solutions are discussed on the basis of the fixed tube model proposed by de Gennes. It is shown that the steady flow viscosity is proportional to M³, where M is the molecular weight, and that the relaxation spectrum exhibits a hump in the relaxation time proportional to M³.     

Molecular properties and basis set quality: An approach based on information theory

Information theory can be used to evaluate the quality of a wave function by considering its ability to give values for some observables as close as possible to the experimental ones. A new method for improving the quality of a wave function is proposed. This paper deals exclusively with the HF (X ¹∑⁺) molecule.     

A variational method to calculate static electronic properties

Using a coupled cluster form of the wave function, a variational method is formulated for calculation of static properties of any order. Corresponding to an appropriate perturbed hamiltonian H() including the relevant static property, a size consistent functional is set up. In a hierarchical fashion, properties of different orders may be found out using a variational method.     

A variational transition state theory description of periselectivity effects on cycloadditions of ketenes with cyclopentadiene

The reaction of cycloaddition of ketene and cyclopentadiene presents experimentally a competing mechanism where the branching ratio between the Woodward?CHoffmann allowed [4+2] and forbidden [2+2] cycloaddition product is 4.56?at ?20?°C, but because the minimum energy path misses the [2+2] product altogether, it has been claimed to lie beyond the scope of transition state theory. In this paper, a variational transition state theory study on this reaction is presented. It is found that the minimum energy path affording the [4+2] product travels through a potential energy plateau very close to the minimum energy path that describes the interconversion between both cycloaddition products, allowing the transfer between both and the direct formation of the forbidden [2+2] product, in this way acting as a means to circumvent the Woodward?CHooffmann rules. Within the domain of the competitive canonical unified statistical theory, a value for the branching ratio in very good agreement with experiment is computed.     

Photocatalytic properties of graphene-supported titania clusters from density-functional theory

Density-functional theory calculations of (TiO₂)_n clusters (n = 1–5) in the gas phase and adsorbed on pristine graphene as well as graphene quantum dots are presented. The cluster adsorption is found to be dominated by van der Waals forces. The electronic structure and in particular the excitation energies of the bare clusters and the TiO₂/graphene composites are found to vary largely in dependence on the size of the respective constituents. This holds in particular for the energy and the spatial localization of the highest occupied and lowest unoccupied molecular orbitals. In addition to a substantial gap narrowing, a pronounced separation of photoexcited electrons and holes is predicted in some instances. This is expected to prolong the lifetime of photoexcited carriers. Altogether, TiO₂/graphene composites are predicted to be promising photocatalysts with improved electronic and photocatalytic properties compared to bulk TiO₂.     

Spectroscopic properties of porphyrin-like photosensitizers: insights from theory

Electronic absorption spectra of six porphyrin-like photosensitizers, porphyrin, chlorin, bacteriochlorin, pheophytin a, porphyrazin, and texaphyrin, have been calculated within the time-dependent DFT framework (TDDFT) in conjunction with the PBE0 hybrid functional. Energetic and orbital aspects are discussed by comparing systems together so as to assess the best molecules for photodynamic therapy applications. Excitation energies and oscillator strengths are found to be in good agreement with both experimental data and previous theoretical works. In particular, whereas significant discrepancies (0.3 eV) appear for Qx bands, results become more reliable as wavelengths decrease. To elucidate the effect of the local environment, we have taken into account solvation either with explicit water molecules interacting via hydrogen bonds with the system or with a continuum model (C-PCM). The supramolecular approach does not affect spectra, while using C-PCM improves Qx and B band values and strengthens intensities significantly. In both gaseous and aqueous phases, texaphyrin, pheophytin a, and bacteriochlorin Qx bands are found in the 600-800 nm range as expected by experimental works. These data are particularly interesting in the perspective of systematic studies of other photosensitizers and should make experimentalists' works easier.