Wick’s theorem and reconstruction schemes for reduced density matrices

We first obtained a closed form of the Wick’s theorem expressed in Grassman wedge product, which is similar to a binomial expansion. With this new expansion, new reconstruction schemes for reduced density matrices are derived rigorously. The higher order reduced density matrices are systematically decomposed into a sum of the lower order reduced density matrices which could be used to solve the contracted Schr?dinger equation.     

A comparison of geometric methods with other methods for the characterization of density matrices

Some recently developed geometric methods for characterizing the subset of density matrices within the space of Hermitian matrices are compared with methods commonly used for the approximate characterization of reduced density matrices. The decomposition of a density matrix into components in terms of the reducing basis set is compared with decomposition in terms of representations of U(r).     

Perturbative correction for the basis set incompleteness error of complete-active-space self-consistent field

To reduce the basis set incompleteness of the complete-active-space self-consistent field (CASSCF) wave function and energy we develop a second-order perturbation correction due to single excitations to complete set of unoccupied states. Other than the one- and two-electron integrals, only one- and two-particle reduced density matrices are required to compute the correction, denoted as [2](S). Benchmark calculations on prototypical ground-state bond-breaking problems show that only the aug-cc-pVXZ basis is needed with the [2](S) correction to match the accuracy of CASSCF energies of the aug-cc-pV(X+1)Z quality.     

Formation of new halogenothiocarbonylsulfenyl halides, XC(S)SY, through photochemical matrix reactions starting from CS2 and a dihalogen molecule XY (XY=Cl2, Br2, or BrCl)

Isolation of a dihalogen molecule XY (XY=Cl2, Br2, or BrCl) with CS2 in a solid Ar matrix at about 15 K leads, by broad-band UV-vis photolysis (200相似文献   

Closed-form expressions for correlated density matrices: application to dispersive interactions and example of He2

Empirically correlated density matrices of N-electron systems are investigated. Closed-form expressions are derived for the one- and two-electron reduced density matrices from a pairwise correlated wave function. Approximate expressions are then proposed which reflect dispersive interactions between closed-shell centrosymmetric subsystems. Said expressions clearly illustrate the consequences of second-order correlation effects on the reduced density matrices. Application is made to a simple example: the He(2) system. Reduced density matrices are explicitly calculated, correct to second order in correlation, and compared with approximations of independent electrons and independent electron pairs. The models proposed allow for variational calculations of interaction energies and equilibrium distance as well as a clear interpretation of dispersive effects on electron distributions. Both exchange and second order correlation effects are shown to play a critical role on the quality of the results.     

Extrapolation of real-space quantum chemical calculations from finite-size super cells to the ideal infinite system. I. Theory

A method is proposed how to calculate the correct density matrix of an infinite polymeric chain from that of a standard finite supercell calculation. The density matrix of the finite supercell is transformed into k-space for all k-values allowed by the periodic boundary conditions. The k-dependent matrices are then unitarily transformed, with each unitary matrix being represented by a set of complex rotation matrices. It is shown that the corresponding angles can be interpolated and extrapolated toward the zone boundaries in a straghtforward manner and that this extrapolation can be done from any finite supercell with reasonable accuracy. This gives rise to an infinite system density matrix for which all fundamental properties are guaranteed by construction. This infinite system density matrix may be used to construct a corrected density matrix for the finite supercell calculation. © 1994 John Wiley & Sons, Inc.     

The role of model systems in the few-body reduction of the N-fermion problem

The problem of upper and lower ground state energy bounds for many-fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.     

A density matrix-based method for the linear-scaling calculation of dynamic second- and third-order properties at the Hartree-Fock and Kohn-Sham density functional theory levels

A density matrix-based time-dependent self-consistent field (D-TDSCF) method for the calculation of dynamic polarizabilities and first hyperpolarizabilities using the Hartree-Fock and Kohn-Sham density functional theory approaches is presented. The D-TDSCF method allows us to reduce the asymptotic scaling behavior of the computational effort from cubic to linear for systems with a nonvanishing band gap. The linear scaling is achieved by combining a density matrix-based reformulation of the TDSCF equations with linear-scaling schemes for the formation of Fock- or Kohn-Sham-type matrices. In our reformulation only potentially linear-scaling matrices enter the formulation and efficient sparse algebra routines can be employed. Furthermore, the corresponding formulas for the first hyperpolarizabilities are given in terms of zeroth- and first-order one-particle reduced density matrices according to Wigner's (2n+1) rule. The scaling behavior of our method is illustrated for first exemplary calculations with systems of up to 1011 atoms and 8899 basis functions.     

Photochemistry of acetylacetone isolated in parahydrogen matrices upon 266 nm irradiation

The photochemistry of the chelated enol form of acetylacetone (AcAc) was investigated by UV excitation of the S(2) state at 266 nm in parahydrogen matrices, complemented by experiments in neon and normal hydrogen matrices. Infrared (IR) spectroscopy, combined with theoretical calculations, was used to identify the photoproducts. Isomerization towards various non-chelated forms (no intramolecular H-bond) of AcAc is the dominant channel whereas fragmentation is very minor. The isomerization kinetics is monitored by IR spectroscopy. Among the seven non-chelated conformers of AcAc, only three are formed in parahydrogen matrices, whereas four are observed in normal hydrogen matrices. This difference suggests that an active tunnelling process between conformers occurs in parahydrogen but is quenched in normal hydrogen where guest-host interactions are stronger. Fragmentation and isomerization of excited AcAc are discussed in the light of these new data. The role of the intermediate triplet state in the S(2)→ S(0) relaxation is confirmed, as the importance of phonons in the condensed phase.     

Structure of Fukui matrices

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices. Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second‐order reduced density matrix cumulant, that is, the explicit contributions of the many‐body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi‐analytical model to denote the occurrence of correlation effects.     

Electron-pair density relaxation holes

The electron-pair density relaxation hole has been defined as the electron-pair density of the real molecule minus the electron-pair density of a reference system consisting of overlapping, spherically averaged, undeformed atoms, positioned at the molecular nuclear coordinates. We have shown how it can be calculated from one- and two-electron reduced density matrices expanded in a Gaussian type basis set. Analysis of the calculated radial electron-pair density holes, from full configuration interaction one- and two-electron reduce density matrices, for the ground states of the hydrogen molecule, the helium dimer and the lithium and beryllium hydrides reveal that the different types of bonding interactions yield distinctively visually recognizable different topological patterns of the electron-pair density relaxation hole.     

Methanesulfenyl fluoride, CH3SF, a missing link in the family of sulfenyl halides: formation and characterization through the matrix photochemistry of methyl thiofluoroformate, FC(O)SCH3

Matrix-isolation experiments have afforded the means of preparing the hitherto unknown sulfur(II) fluoride, methanesulfenyl fluoride, CH3SF. Broadband UV-visible irradiation of methyl thiofluoroformate, FC(O)SCH3, isolated in a solid Ar matrix results, first, in photoisomerization of the syn into the anti form of the molecule, and, subsequently, in the elimination of CO with the concomitant formation of CH3SF. Continued irradiation brings about tautomerization of this product with the detachment and migration of a hydrogen from the methyl group to give the molecular complex H2C==S...HF. The changes have been monitored and the photoproducts detected and identified by the IR spectra of the matrices, and the conclusions confirmed: 1) with reference to the corresponding behavior of the perdeuterated molecule FC(O)SCD3; 2) by analogy with the properties of related molecules, for example, ClC(O)SCH3, CH3SCl, and H2C==S...HCl, and; 3) by comparison with the vibrational properties simulated for the different molecules by ab initio and density functional theory methods.     

Semigroups and the density matrix formulation of quantum mechanics

A formulation of quantum mechanics is presented based on the theory of semigroups and the associated enveloping algebras of functions defined on countable subsemigroups. The existence of a unique *-involution is not assumed. The fundamental elements of a semigroup are identified with experimental precedures for the separation of subensembles from a given ensemble of experimental systems. Observables are represented as elements of enveloping algebras, and ensembles as density matrices within an enveloping algebra. The statistical properties of ensembles are expressed in terms of traces defined on the semigroup and its enveloping algebras. The elements and generators of the Poincaré group can be defined and interpreted in the usual way. A variety of applications is described, in which the theory of the density matrix plays an essential or effective role. Advantages associated with the resulting freedom from the limitations of Hilbert space are illustrated.     

Geometry of density matrices. VI. Superoperators and unitary invariance

We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.     

A DFT conformational analysis and VCD study on methyl tetrahydrofuran-2-carboxylate

DFT calculations were performed on (S)-methyl tetrahydrofuran-2-carboxylate to facilitate the interpretation of IR and VCD spectra. The potential energy surface could not be described unambiguously using the 6-31G* basis set in combination with different density functionals including B1LYP, B3LYP, B3P86, B3PW91, B98, BHandH, BHandHLYP, MPW1PW91 and PBE1PBE. In contrast, a uniform conformational picture could be found using the cc-pVTZ basis set. Using this large basis set and the collection of nine functionals from above, the dipole and rotational strengths were calculated, and compared to experimental values which were extracted from the experimental IR and VCD spectra for (+)-(S)-methyl tetrahydrofuran-2-carboxylate. A detailed analysis on the agreement between experiment and simulated spectra was performed by assigning the experimental bands based on the harmonic fundamentals obtained for all functionals except BHandH, which performs badly over the whole line. Assessing the dipole strengths, all tested functionals perform equally well. For the rotational strengths, differences can be observed: B3LYP, B1LYP and B98 give the highest correlation with experiment, while PBE1PBE gives the lowest correlation. Comparable conclusions are obtained using a neighborhood similarity measure.     

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.     

Decomposition of the first-order reduced density matrix: an isopycnic localization treatment

In this work, we propose a partitioning of the first-order reduced density matrix corresponding to an N-electron system into first-order reduced density matrices associated with regions defined in the real space (regional matrices). The treatment is based on an isopycnic orbital localization transformation that provides regional matrices that are diagonalized by identical localized orbitals, having many attributes associated with chemical concepts (appropriate localization in space, high transferability, etc.). Although the obtained numerical values are similar to those arising from previous studies, their interpretation is more rigorous and the computational cost is much lower.     

Infrared spectroscopic studies of free-base tetraphenylporphine and its dication

We present here the infrared absorption spectra of free-base tetraphenylporphine and its dication. Most of the allowed IR bands of porphyrin skeletal are observed in pairs due to two-fold symmetry of the free-base tetraphenylporphine. Observation of some new bands, disappearance of few bands in the IR spectrum of dication are interpreted on the basis of point group symmetry S4. Intensity change in the observed bands due to vibrational motion of the phenyl rings for dication is also explained on the basis of symmetry of dication. Sharing of electrons of the B(1u) orbitals by the two added protons are responsible for the shifts in the position of certain IR bands for dication.     

A new partitioning scheme for molecular interacting systems within a multiconfigurational or monoconfigurational Hartree–Fock formalism

A new method, based on the spatial decomposition of the reduced‐density and pair‐density matrices and the indistinguishable integrals formalism, is introduced to partition the molecular and stabilization energies into meaningful fragments. These are defined as entirely flexible variable‐size entities, for example, atoms, group of atoms, ions, and interacting monomers. This new partitioning scheme is especially appropriated to study systems in which a directly bonded group‐transfer process occurs. In these cases, the stabilization energies are partitioned into an intrafragment component, associated with the difference of intrinsic affinity to the transferred group between the involved fragments, and an interfragment component, associated with the difference of the magnitude of the interaction between the fragments in the initial and final binding complexes. This method was applied to the study of the arginine–carboxylate interactions, allowing us to have insight into what really happens in this system. Two (zwitterionic and neutral) binding complexes can be considered. The main effects accountable for the preferential stabilizations of the binding complexes are determined to be basis‐set independent. The zwitterionic complex is favored by the interfragment component, while the neutral complex is favored by the larger intrinsic proton affinity of the acetate relatively to the methylguanidium. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 157–176, 1999     

Hierarchy equations for reduced density matrices in different representations

We derive exact relationships for the reduced density matrices in representations where the transformation matrix is a product of one-body transformation matrices. We specialize to the momentum and onebody energy representations. By decoupling the equations we are able to write the Hartree-Fock equation in terms of the first-order density matrix in an arbitrary representation. Applications to reduced local energy and the correlation problem are discussed.