arXiv subjects

A. I. Panin

Publications and source records attributed to A. I. Panin.

8 recordsLinked to original sources

arXiv2007

Methods of Modern Differential Geometry in Quantum Chemistry: TD Theories on Grassmann and Hartree-Fock Manifolds

Hamiltonian and Schrodinger evolution equations on finite-dimensional projective space are analyzed in detail. Hartree-Fock (HF) manifold is introduced as a submanifold of many electron projective space of states. Evolution equations, exact and linearized, on this manifold are studied. Comparison of matrices of linearized Schrodinger equations on many electron projective space and on the corresponding HF manifold reveals the appearance in the HF case a constraining matrix that involves matrix elements of many-electron Hamiltonian between HF state and double excited determinants. Character of dependence of transition energies on the matrix elements of constraining matrix is established by means of perturbation analysis. It is demonstrated that success of time-dependent HF theory in calculation of transition energies is mainly due to the wrong behavior of these energies as functions of matrix elements of constraining matrix as compared with the exact transition energies

physics.chem-ph↗

arXiv2006

1-Density Operators and Algebraic Version of The Hohenberg-Kohn Theorem

Interrelation of the Coleman's representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman's set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman's polyhedrons is obtained.

physics.chem-ph↗

arXiv2005

Electronic Fock space as associative superalgebra

New algebraic structure on electronic Fock space is studied in detail. This structure is defined in terms of a certain multiplication of many electron wave functions and has close interrelation with coupled cluster and similar approaches. Its study clarifies and simplifies the mathematical backgrounds of these approaches. And even more, it leads to many relations that would be very difficult to derive using conventional technique. Formulas for action of the creation-annihilation operators on products of state vectors are derived. Explicit expressions for action of simplest particle-conserving products of the creation-annihilation operators on powers of state vectors are given. General scheme of parametrization of representable density operators of arbitrary order is presented.

physics.chem-ph↗

arXiv2005

Electronic Fock spaces: Phase prefactors and hidden symmetry

Efficient technique of manipulation with phase prefactors in electronic Fock spaces is developed. Its power is demonstrated on example of both relatively simple classic configuration interaction matrix element evaluation and essentially more complicated coupled cluster case. Interpretation of coupled cluster theory in terms of a certain commutative algebra is given.

physics.chem-ph↗

arXiv2003

Restoration of Many Electron Wave Functions from One-Electron Density

General theorem describing a relation between diagonal of one-electron density matrix and a certain class of many-electron ensembles of determinant states is proved. As a corollary to this theorem a constructive proof of sufficiency of Coleman's representability conditions is obtained. It is shown that there exist rigorous schemes for construction of energy of many-electron system as functionals of one-electron density.

physics.chem-ph↗

arXiv2002

(p,q)-Sheaves and the Representability Problem

General properties of new models of the electronic Fock spaces based on the notion of (p,q)-sheaves are studied. Interrelation between simple sheaves and density operators is established. Explicit expressions for the transformed reduced Hamiltonians in terms of the standard creation-annihilation operators are presented. General scheme of parametrization of p-electron states by k-electron means (k=2,3,...) is described and studied in detail for the case of sheaves induced by k-electron wavefunctions. It is demonstrated that under certain conditions p-electron problem may be reformulated as the eigenvalue problem in k-electron space equipped with certain p-electron metric. Simple numerical examples are given to illustrate our approach.

physics.chem-ph↗

arXiv2001

Pure Representability Problem and New Models of the Electronic Fock Space

New models of the Fock space sector corresponding to some fixed number of electrons are introduced. These models originate from the representability theory and their practical implementation may lead to essential reduction of dimensions of intermediate Configuration Interaction spaces. A certain zero-order theory that gives wave functions approximately equivalent to ones obtained by accounting all excitations from the Hartree-Fock reference state up to the q-th order is proposed. Simple numerical examples are given to illustrate the approach.

physics.chem-ph↗

arXiv2001

Approximate Solution of the Representability Problem

Approximate solution of the ensemble representability problem for density operators of arbitrary order is obtained. This solution is closely related to the ``Q condition'' of A.J.Coleman. The representability conditions are formulated in orbital representation and are easy to use. They are tested numerically on the base of CI calculation of simple atomic and molecular systems. General scheme of construction of the contraction operator right inverses is proposed and the explicit expression for the right inverse associated with the expansion operator is derived as an example. Two algorithms for direct 2-density matrix determination are described.

physics.chem-ph↗