We present a method for including the geometric phase in quantum reactive scattering computations... more We present a method for including the geometric phase in quantum reactive scattering computations based on the log derivative version of the Kohn variational principle. A new variational functional is developed which includes the influence of the geometric phase through modifications in the momentum operators. The system investigated is a two-dimensional reactive scattering model which includes the vector potential induced by the magnetic field of an infinitely long solenoid. The coordinates used in this model are analogous to Jacobi coordinates used in atom–diatom systems. Some interesting features of this study include the gauge invariance of the scattering probabilities, symmetry adaptation of the wave function, and the behavior of the probability density in the presence of the geometric phase.
The log derivative version of the Kohn variational principle is reviewed in the context of a gene... more The log derivative version of the Kohn variational principle is reviewed in the context of a general bimolecular chemical reaction. The basis of this review, namely, the Wigner and Eisenbud general formulation of rearrangement scattering, has been well known for many years. Therefore, so as to avoid any unnecessary confusion, the relationship between their equally famous ℛ matrix theory and Kohn’s variational derivation is carefully described. The log derivative matrix is then eliminated from a basis set representation of Kohn’s principle to leave a unitary and symmetric variational expression for the scattering matrix S. This new expression is expected to find its most fruitful application in the iterative solution of very large quantum scattering problems for which transitions from only a few initial states are required.
In the variational subspace valence bond (VSVB) [G. D. Fletcher, J. Chem. Phys. 142, 134112 (2015... more In the variational subspace valence bond (VSVB) [G. D. Fletcher, J. Chem. Phys. 142, 134112 (2015)] method, the electronic orbitals comprising the wave function correspond to chemically meaningful objects, such as bonds, lone pairs, atomic cores, and so on. Selected regions of a molecule (for example, a single chemical bond, as opposed to all the bonds) can be modeled with different levels of basis set and possible methods for modeling correlation from the other regions. The interactions between the components of a molecule (say, a bond and a neighboring orbital) can then be studied in detail for their impact on a chemical phenomenon while avoiding the expense of necessarily applying the higher levels and methods to the entire molecule. This work presents the theoretical basis for modeling correlation effects between specific electron pairs by incorporating terms in the inter-electronic coordinates (“r12”) into VSVB. The approach is validated with calculations on small systems using single-reference wave functions.
The Kohn variational principle for the log-derivative matrix is used to calculate integral cross ... more The Kohn variational principle for the log-derivative matrix is used to calculate integral cross sections for H+D2 (v=0, j=0) to D+HD (v′=0,1,2, all j′) at the experimentally accessible collision energies of 0.55 and 1.3 eV. Comparison is made with experimental and theoretical studies in the literature. Product state relative rotational distributions, vibrational branching ratios, and energy partitioning fractions are all in good agreement with the recent experimental results of Rinnen, Kliner, and Zare. Absolute cross sections are compared with the experimental work of Levene et al. and Johnson et al. Our results agree very well with their experiments. It is found that the quasiclassical results of Blais and Truhlar compare well with the present exact quantum mechanical predictions in many respects, however, the product rotational distributions are ‘‘hotter’’ than the quantal ones.
The hydrogen exchange reaction in its H+D2(v=0,j=0)→HD(v′=0,j′)+D isotopic variant has been inves... more The hydrogen exchange reaction in its H+D2(v=0,j=0)→HD(v′=0,j′)+D isotopic variant has been investigated theoretically and experimentally at the collision energies 0.52 eV, 0.531 eV and 0.54 eV. A detailed comparison of converged quantum mechanical scattering calculations and state-to-state molecular beam experiments has allowed a direct assessment of the quality of the different ab initio potential energy surfaces used in the calculations, and strongly favors the newly refined version of the Boothroyd–Keogh–Martin–Peterson surface. The differences found in the calculations are traced back to slight differences in the topology of the potential energy surfaces.
A new translational basis set is introduced for quantum reactive scattering calculations that use... more A new translational basis set is introduced for quantum reactive scattering calculations that use the log derivative version of the Kohn variational principle. This basis set, which is similar in many respects to that used in electron–atom scattering calculations by Burke and Robb, is obtained by contracting a primitive basis of Lobatto shape functions to the box eigenfunctions of a one-dimensional reference Hamiltonian H0. In addition, a single energy-dependent scattering function is included in the variational expansion to ensure completeness at the boundary of the box. One fairly obvious choice for the reference Hamiltonian in an atom–diatom reaction is suggested, and all of the equations which are actually needed to implement the method in this context are carefully described. Example applications to the three-dimensional F+H2 reaction are then chosen to illustrate the practical potential of the approach.
We present a method for including the geometric phase in quantum reactive scattering computations... more We present a method for including the geometric phase in quantum reactive scattering computations based on the log derivative version of the Kohn variational principle. A new variational functional is developed which includes the influence of the geometric phase through modifications in the momentum operators. The system investigated is a two-dimensional reactive scattering model which includes the vector potential induced by the magnetic field of an infinitely long solenoid. The coordinates used in this model are analogous to Jacobi coordinates used in atom–diatom systems. Some interesting features of this study include the gauge invariance of the scattering probabilities, symmetry adaptation of the wave function, and the behavior of the probability density in the presence of the geometric phase.
The log derivative version of the Kohn variational principle is reviewed in the context of a gene... more The log derivative version of the Kohn variational principle is reviewed in the context of a general bimolecular chemical reaction. The basis of this review, namely, the Wigner and Eisenbud general formulation of rearrangement scattering, has been well known for many years. Therefore, so as to avoid any unnecessary confusion, the relationship between their equally famous ℛ matrix theory and Kohn’s variational derivation is carefully described. The log derivative matrix is then eliminated from a basis set representation of Kohn’s principle to leave a unitary and symmetric variational expression for the scattering matrix S. This new expression is expected to find its most fruitful application in the iterative solution of very large quantum scattering problems for which transitions from only a few initial states are required.
In the variational subspace valence bond (VSVB) [G. D. Fletcher, J. Chem. Phys. 142, 134112 (2015... more In the variational subspace valence bond (VSVB) [G. D. Fletcher, J. Chem. Phys. 142, 134112 (2015)] method, the electronic orbitals comprising the wave function correspond to chemically meaningful objects, such as bonds, lone pairs, atomic cores, and so on. Selected regions of a molecule (for example, a single chemical bond, as opposed to all the bonds) can be modeled with different levels of basis set and possible methods for modeling correlation from the other regions. The interactions between the components of a molecule (say, a bond and a neighboring orbital) can then be studied in detail for their impact on a chemical phenomenon while avoiding the expense of necessarily applying the higher levels and methods to the entire molecule. This work presents the theoretical basis for modeling correlation effects between specific electron pairs by incorporating terms in the inter-electronic coordinates (“r12”) into VSVB. The approach is validated with calculations on small systems using single-reference wave functions.
The Kohn variational principle for the log-derivative matrix is used to calculate integral cross ... more The Kohn variational principle for the log-derivative matrix is used to calculate integral cross sections for H+D2 (v=0, j=0) to D+HD (v′=0,1,2, all j′) at the experimentally accessible collision energies of 0.55 and 1.3 eV. Comparison is made with experimental and theoretical studies in the literature. Product state relative rotational distributions, vibrational branching ratios, and energy partitioning fractions are all in good agreement with the recent experimental results of Rinnen, Kliner, and Zare. Absolute cross sections are compared with the experimental work of Levene et al. and Johnson et al. Our results agree very well with their experiments. It is found that the quasiclassical results of Blais and Truhlar compare well with the present exact quantum mechanical predictions in many respects, however, the product rotational distributions are ‘‘hotter’’ than the quantal ones.
The hydrogen exchange reaction in its H+D2(v=0,j=0)→HD(v′=0,j′)+D isotopic variant has been inves... more The hydrogen exchange reaction in its H+D2(v=0,j=0)→HD(v′=0,j′)+D isotopic variant has been investigated theoretically and experimentally at the collision energies 0.52 eV, 0.531 eV and 0.54 eV. A detailed comparison of converged quantum mechanical scattering calculations and state-to-state molecular beam experiments has allowed a direct assessment of the quality of the different ab initio potential energy surfaces used in the calculations, and strongly favors the newly refined version of the Boothroyd–Keogh–Martin–Peterson surface. The differences found in the calculations are traced back to slight differences in the topology of the potential energy surfaces.
A new translational basis set is introduced for quantum reactive scattering calculations that use... more A new translational basis set is introduced for quantum reactive scattering calculations that use the log derivative version of the Kohn variational principle. This basis set, which is similar in many respects to that used in electron–atom scattering calculations by Burke and Robb, is obtained by contracting a primitive basis of Lobatto shape functions to the box eigenfunctions of a one-dimensional reference Hamiltonian H0. In addition, a single energy-dependent scattering function is included in the variational expansion to ensure completeness at the boundary of the box. One fairly obvious choice for the reference Hamiltonian in an atom–diatom reaction is suggested, and all of the equations which are actually needed to implement the method in this context are carefully described. Example applications to the three-dimensional F+H2 reaction are then chosen to illustrate the practical potential of the approach.
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Papers by Michael D'Mello