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Lev Kantorovich - One of the best experts on this subject based on the ideXlab platform.

  • Arrow Diagram theory for non orthogonal electronic groups the continued fractions method
    Journal of Physics: Condensed Matter, 2009
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    The group function theory by Tolpygo and McWeeny is a useful tool in treating quantum systems that can be represented as a set of localized electronic groups (e.g. atoms, molecules or bonds). It provides a general means of taking into account intra-correlation effects inside the groups without assuming that the interaction between the groups is weak. For non-orthogonal group functions the Arrow Diagram (AD) technique provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators which are needed, for example, for calculating the total energy of the system or its electron density. The total wavefunction of the system is represented as an antisymmetrized product of non-orthogonal electron group functions ΦI of each group I in the system. However, application of the AD theory to extended (e.g. infinite) systems (such as biological molecules or crystals) is not straightforward, since the calculation of the mean value of an operator requires that each term of the Diagram expansion be divided by the normalization integral S = Ψ|Ψ which is given by an AD expansion as well. In our previous work, we cast the mean value of a symmetrical operator in the form of an AD expansion which is a linear combination of linked (connected) ADs multiplied by numerical pre-factors. To obtain the pre-factors, a method based on power series expansion with respect to overlap was developed and tested for a simple 1D Hartree–Fock (HF) ring model. In the present paper this method is first tested on a 2D HF model, and we find that the power series expansion for the pre-factors converges extremely slowly to the exact solution. Instead, we suggest another, more powerful, method based on a continued fraction expansion of the pre-factors that approaches the exact solution much faster. The method is illustrated on the calculation of the electron density for the 2D HF model. It provides a powerful technique for treating extended systems consisting of a large number of strongly localized electronic groups.

  • Arrow Diagram approach to nonorthogonal electron group functions in extended systems
    Journal of Physics: Condensed Matter, 2006
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    For nonorthogonal electron group functions, the Arrow Diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators . The total wavefunction of the system is represented as an antisymmetrized product of nonorthogonal many-electron group functions ΦI of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the Diagram expansion be divided by the normalization integral which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree–Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).

  • Arrow Diagram method based on overlapping electronic groups corrections to the linked ad theorem
    arXiv: Chemical Physics, 2004
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    Arrow Diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $ $, of symmetrical operators, $\hat{O}$, in quantum chemistry when the total system wavefunction $\Psi$ is represented as an antisymmetrised product of overlapping many-electron group functions, $\Phi_{A}$, corresponding to each part (group) $A$ of the system: $\Psi=\hat{A}\prod_{A}\Phi_{A}$. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the Diagram expansion is to be divided by the normalisation integral $S= $, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.

  • Diagram technique for nonorthogonal electron group functions ii reduced density matrices and total energy
    Journal of Chemical Physics, 1992
    Co-Authors: Lev Kantorovich, B P Zapol
    Abstract:

    In part I, both the Arrow Diagram (AD) and expanded AD decompositions of the antisymmetrization operator A for an N‐electron system with wave function represented by the product of mutually nonorthogonal group functions have been considered. Based on them, new Diagrams for decompositions of normalization and overlap integrals, reduced density matrices, as well as for total electronic energy of the system are proposed and discussed in detail in the present part. The rules for evaluation of the contribution of each Diagram in the form of an analytical expression are obtained. Both the strong and p‐orthogonality approximations are discussed.

Arnaud Mortier - One of the best experts on this subject based on the ideXlab platform.

B P Zapol - One of the best experts on this subject based on the ideXlab platform.

  • Diagram technique for nonorthogonal electron group functions ii reduced density matrices and total energy
    Journal of Chemical Physics, 1992
    Co-Authors: Lev Kantorovich, B P Zapol
    Abstract:

    In part I, both the Arrow Diagram (AD) and expanded AD decompositions of the antisymmetrization operator A for an N‐electron system with wave function represented by the product of mutually nonorthogonal group functions have been considered. Based on them, new Diagrams for decompositions of normalization and overlap integrals, reduced density matrices, as well as for total electronic energy of the system are proposed and discussed in detail in the present part. The rules for evaluation of the contribution of each Diagram in the form of an analytical expression are obtained. Both the strong and p‐orthogonality approximations are discussed.

Yu Wang - One of the best experts on this subject based on the ideXlab platform.

  • Arrow Diagram theory for non orthogonal electronic groups the continued fractions method
    Journal of Physics: Condensed Matter, 2009
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    The group function theory by Tolpygo and McWeeny is a useful tool in treating quantum systems that can be represented as a set of localized electronic groups (e.g. atoms, molecules or bonds). It provides a general means of taking into account intra-correlation effects inside the groups without assuming that the interaction between the groups is weak. For non-orthogonal group functions the Arrow Diagram (AD) technique provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators which are needed, for example, for calculating the total energy of the system or its electron density. The total wavefunction of the system is represented as an antisymmetrized product of non-orthogonal electron group functions ΦI of each group I in the system. However, application of the AD theory to extended (e.g. infinite) systems (such as biological molecules or crystals) is not straightforward, since the calculation of the mean value of an operator requires that each term of the Diagram expansion be divided by the normalization integral S = Ψ|Ψ which is given by an AD expansion as well. In our previous work, we cast the mean value of a symmetrical operator in the form of an AD expansion which is a linear combination of linked (connected) ADs multiplied by numerical pre-factors. To obtain the pre-factors, a method based on power series expansion with respect to overlap was developed and tested for a simple 1D Hartree–Fock (HF) ring model. In the present paper this method is first tested on a 2D HF model, and we find that the power series expansion for the pre-factors converges extremely slowly to the exact solution. Instead, we suggest another, more powerful, method based on a continued fraction expansion of the pre-factors that approaches the exact solution much faster. The method is illustrated on the calculation of the electron density for the 2D HF model. It provides a powerful technique for treating extended systems consisting of a large number of strongly localized electronic groups.

  • Arrow Diagram approach to nonorthogonal electron group functions in extended systems
    Journal of Physics: Condensed Matter, 2006
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    For nonorthogonal electron group functions, the Arrow Diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators . The total wavefunction of the system is represented as an antisymmetrized product of nonorthogonal many-electron group functions ΦI of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the Diagram expansion be divided by the normalization integral which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree–Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).

  • Arrow Diagram method based on overlapping electronic groups corrections to the linked ad theorem
    arXiv: Chemical Physics, 2004
    Co-Authors: Yu Wang, Lev Kantorovich
    Abstract:

    Arrow Diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $ $, of symmetrical operators, $\hat{O}$, in quantum chemistry when the total system wavefunction $\Psi$ is represented as an antisymmetrised product of overlapping many-electron group functions, $\Phi_{A}$, corresponding to each part (group) $A$ of the system: $\Psi=\hat{A}\prod_{A}\Phi_{A}$. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the Diagram expansion is to be divided by the normalisation integral $S= $, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.

Aynur Ozdas - One of the best experts on this subject based on the ideXlab platform.

  • the effect of Arrow Diagrams on achievement in applying the chain rule
    PRIMUS, 2007
    Co-Authors: Tangul Uygur, Aynur Ozdas
    Abstract:

    Abstract In this study the effectiveness of an Arrow Diagram which can help students apply the Chain Rule was investigated. Different variations of this Diagram were used as mnemonic devices for applying the Chain Rule. For the investigation two instruments were developed, diagnostic test and post-test. The diagnostic test was developed to determine the students' difficulties with the Chain Rule and to create matched groups. It was administered to 76 students taking the Advanced Calculus Course. By matching according to the results of the diagnostic test, the sample of 24 pairs of subjects is obtained. The results of the post-test, administered after the teaching program, indicated that the Arrow Diagram had positive effects on applying the Chain Rule.