The Experts below are selected from a list of 90 Experts worldwide ranked by ideXlab platform
Friedrich Grein - One of the best experts on this subject based on the ideXlab platform.
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quadrupole octopole and hexadecapole electric moments of σ π δ and φ electronic states cylindrically asymmetric charge density distributions in linear molecules with Nonzero electronic angular momentum
Journal of Chemical Physics, 2007Co-Authors: Pablo J Bruna, Friedrich GreinAbstract:The number of independent components, n, of traceless electric 2l-multipole moments is deTermined for C∞v molecules in Σ±, Π, Δ, and Φ electronic states (Λ=0,1,2,3). Each 2l pole is defined by a rank-l irreducible tensor with (2l+1) components Pm(l) proportional to the solid spherical harmonic rlYml(θ,φ). Here we focus our attention on 2l poles with l=2,3,4 (quadrupole Θ, octopole Ω, and hexadecapole Φ). An important conclusion of this study is that n can be 1 or 2 depending on both the multipole rank l and state quantum number Λ. For Σ±(Λ=0) states, all 2l poles have one independent parameter (n=1). For spatially degenerate states—Π, Δ, and Φ (Λ=1,2,3)—the general rule reads n=1 for l<2∣Λ∣ (when the 2l-pole rank lies below 2∣Λ∣) but n=2 for higher 2l poles with l⩾2∣Λ∣. The second Nonzero Term is the off-diagonal matrix element ⟨ψ+Λ∣P∣m∣=2Λ(l)∣ψ−Λ⟩. Thus, a Π(Λ=1) state has one dipole (μz) but two independent 2l poles for l⩾2—starting with the quadrupole [Θzz,(Θxx−Θyy)]. A Δ(Λ=2) state has n=1 for 2(1,2,3...
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quadrupole octopole and hexadecapole electric moments of σ π δ and φ electronic states cylindrically asymmetric charge density distributions in linear molecules with Nonzero electronic angular momentum
Journal of Chemical Physics, 2007Co-Authors: Pablo J Bruna, Friedrich GreinAbstract:The number of independent components, n, of traceless electric 2(l)-multipole moments is deTermined for C(infinity v) molecules in Sigma(+/-), Pi, Delta, and Phi electronic states (Lambda=0,1,2,3). Each 2(l) pole is defined by a rank-l irreducible tensor with (2l+1) components P(m)((l)) proportional to the solid spherical harmonic r(l)Y(m)(l)(theta,phi). Here we focus our attention on 2(l) poles with l=2,3,4 (quadrupole Theta, octopole Omega, and hexadecapole Phi). An important conclusion of this study is that n can be 1 or 2 depending on both the multipole rank l and state quantum number Lambda. For Sigma(+/-)(Lambda=0) states, all 2(l) poles have one independent parameter (n=1). For spatially degenerate states--Pi, Delta, and Phi (Lambda=1,2,3)--the general rule reads n=1 for l or=2/Lambda/. The second Nonzero Term is the off-diagonal matrix element [formula: see text]. Thus, a Pi(Lambda=1) state has one dipole (mu(z)) but two independent 2(l) poles for l>or=2--starting with the quadrupole [Theta(zz),(Theta(xx)-Theta(yy))]. A Delta(Lambda=2) state has n=1 for 2((1,2,3)) poles (mu(z),Theta(zz),Omega(zzz)) but n=2 for higher 2((l>or=4)) poles--from the hexadecapole Phi up. For Phi(Lambda=3) states, it holds that n=1 for 2(1) to 2(5) poles but n=2 for all 2((l>or=6)) poles. In short, what is usually stated in the literature--that n=1 for all possible 2(l) poles of linear molecules--only applies to Sigma(+/-) states. For degenerate states with n=2, all Cartesian 2(l)-pole components (l>or=2/Lambda/) can be expressed as linear combinations of two irreducible multipoles, P(m=0)((l)) and P/m/=2 Lambda)((l)) [parallel (z axis) and anisotropy (xy plane)]. Our predictions are exemplified by the Theta, Omega, and Phi moments calculated for Lambda=0-3 states of selected diatomics (in parentheses): X (2)Sigma(+)(CN), X (2)Pi(NO), a (3)Pi(u)(C(2)), X (2)Delta(NiH), X (3)Delta(TiO), X (3)Phi(CoF), and X (4)Phi(TiF). States of Pi symmetry are most affected by the deviation from axial symmetry.
Pablo J Bruna - One of the best experts on this subject based on the ideXlab platform.
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quadrupole octopole and hexadecapole electric moments of σ π δ and φ electronic states cylindrically asymmetric charge density distributions in linear molecules with Nonzero electronic angular momentum
Journal of Chemical Physics, 2007Co-Authors: Pablo J Bruna, Friedrich GreinAbstract:The number of independent components, n, of traceless electric 2l-multipole moments is deTermined for C∞v molecules in Σ±, Π, Δ, and Φ electronic states (Λ=0,1,2,3). Each 2l pole is defined by a rank-l irreducible tensor with (2l+1) components Pm(l) proportional to the solid spherical harmonic rlYml(θ,φ). Here we focus our attention on 2l poles with l=2,3,4 (quadrupole Θ, octopole Ω, and hexadecapole Φ). An important conclusion of this study is that n can be 1 or 2 depending on both the multipole rank l and state quantum number Λ. For Σ±(Λ=0) states, all 2l poles have one independent parameter (n=1). For spatially degenerate states—Π, Δ, and Φ (Λ=1,2,3)—the general rule reads n=1 for l<2∣Λ∣ (when the 2l-pole rank lies below 2∣Λ∣) but n=2 for higher 2l poles with l⩾2∣Λ∣. The second Nonzero Term is the off-diagonal matrix element ⟨ψ+Λ∣P∣m∣=2Λ(l)∣ψ−Λ⟩. Thus, a Π(Λ=1) state has one dipole (μz) but two independent 2l poles for l⩾2—starting with the quadrupole [Θzz,(Θxx−Θyy)]. A Δ(Λ=2) state has n=1 for 2(1,2,3...
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quadrupole octopole and hexadecapole electric moments of σ π δ and φ electronic states cylindrically asymmetric charge density distributions in linear molecules with Nonzero electronic angular momentum
Journal of Chemical Physics, 2007Co-Authors: Pablo J Bruna, Friedrich GreinAbstract:The number of independent components, n, of traceless electric 2(l)-multipole moments is deTermined for C(infinity v) molecules in Sigma(+/-), Pi, Delta, and Phi electronic states (Lambda=0,1,2,3). Each 2(l) pole is defined by a rank-l irreducible tensor with (2l+1) components P(m)((l)) proportional to the solid spherical harmonic r(l)Y(m)(l)(theta,phi). Here we focus our attention on 2(l) poles with l=2,3,4 (quadrupole Theta, octopole Omega, and hexadecapole Phi). An important conclusion of this study is that n can be 1 or 2 depending on both the multipole rank l and state quantum number Lambda. For Sigma(+/-)(Lambda=0) states, all 2(l) poles have one independent parameter (n=1). For spatially degenerate states--Pi, Delta, and Phi (Lambda=1,2,3)--the general rule reads n=1 for l or=2/Lambda/. The second Nonzero Term is the off-diagonal matrix element [formula: see text]. Thus, a Pi(Lambda=1) state has one dipole (mu(z)) but two independent 2(l) poles for l>or=2--starting with the quadrupole [Theta(zz),(Theta(xx)-Theta(yy))]. A Delta(Lambda=2) state has n=1 for 2((1,2,3)) poles (mu(z),Theta(zz),Omega(zzz)) but n=2 for higher 2((l>or=4)) poles--from the hexadecapole Phi up. For Phi(Lambda=3) states, it holds that n=1 for 2(1) to 2(5) poles but n=2 for all 2((l>or=6)) poles. In short, what is usually stated in the literature--that n=1 for all possible 2(l) poles of linear molecules--only applies to Sigma(+/-) states. For degenerate states with n=2, all Cartesian 2(l)-pole components (l>or=2/Lambda/) can be expressed as linear combinations of two irreducible multipoles, P(m=0)((l)) and P/m/=2 Lambda)((l)) [parallel (z axis) and anisotropy (xy plane)]. Our predictions are exemplified by the Theta, Omega, and Phi moments calculated for Lambda=0-3 states of selected diatomics (in parentheses): X (2)Sigma(+)(CN), X (2)Pi(NO), a (3)Pi(u)(C(2)), X (2)Delta(NiH), X (3)Delta(TiO), X (3)Phi(CoF), and X (4)Phi(TiF). States of Pi symmetry are most affected by the deviation from axial symmetry.
Dmitry Novikov - One of the best experts on this subject based on the ideXlab platform.
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Godbillon–Vey sequence and Françoise algorithm
Bulletin des Sciences Mathématiques, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:We consider foliations given by deformations dF + epsilon omega of exact forms dF in C-2 in a neighborhood of a family of cycles -gamma(t) subset of F-1(t). In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function Delta along gamma of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation of integrability of a form omega. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.
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Godbillon-Vey sequence and Francoise algorithm
arXiv: Dynamical Systems, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:We consider foliations given by deformations $dF+\epsilon\omega$ of exact forms $dF$ in $\mathbb{C}^2$ in a neighborhood of a family of cycles $\gamma(t)\subset F^{-1}(t)$. In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function $\Delta$ along $\gamma$ of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation integrability of a form $\omega$. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.
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Godbillon–Vey sequence and Françoise algorithm
Bulletin des Sciences Mathématiques, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:Abstract We consider foliations given by deformations d F + ϵ ω of exact forms dF in C 2 in a neighborhood of a family of cycles γ ( t ) ⊂ F − 1 ( t ) . In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function Δ along γ of such deformations. This algorithm recalls the well-known Godbillon–Vey sequences discovered in 1971 for investigation of integrability of a form ω. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon–Vey sequences to the Francoise algorithm settings.
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bounding the length of iterated integrals of the first Nonzero melnikov function
Moscow Mathematical Journal, 2018Co-Authors: Pavao Mardesic, Dmitry Novikov, Laura Ortizbobadilla, Jessie PontigoherreraAbstract:We consider small polynomial deformations of integrable systems of the form $dF=0, F\in\mathbb{C}[x,y]$ and the first Nonzero Term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ epends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first Nonzero Term $\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.
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bounding the length of iterated integrals of the first Nonzero melnikov function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Pavao Mardesic, Dmitry Novikov, Laura Ortizbobadilla, Jessie PontigoherreraAbstract:We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb{C}[x,y]$ and the first Nonzero Term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ depends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first Nonzero Term $M_\mu$ depending only on the topology of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral i.e. by an iterated integral of length $1$. We conjecture that our bound is optimal.
Jessie Pontigo-herrera - One of the best experts on this subject based on the ideXlab platform.
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Godbillon–Vey sequence and Françoise algorithm
Bulletin des Sciences Mathématiques, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:We consider foliations given by deformations dF + epsilon omega of exact forms dF in C-2 in a neighborhood of a family of cycles -gamma(t) subset of F-1(t). In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function Delta along gamma of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation of integrability of a form omega. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.
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Godbillon-Vey sequence and Francoise algorithm
arXiv: Dynamical Systems, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:We consider foliations given by deformations $dF+\epsilon\omega$ of exact forms $dF$ in $\mathbb{C}^2$ in a neighborhood of a family of cycles $\gamma(t)\subset F^{-1}(t)$. In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function $\Delta$ along $\gamma$ of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation integrability of a form $\omega$. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.
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Godbillon–Vey sequence and Françoise algorithm
Bulletin des Sciences Mathématiques, 2019Co-Authors: Pavao Mardešić, Dmitry Novikov, Laura Ortiz-bobadilla, Jessie Pontigo-herreraAbstract:Abstract We consider foliations given by deformations d F + ϵ ω of exact forms dF in C 2 in a neighborhood of a family of cycles γ ( t ) ⊂ F − 1 ( t ) . In 1996 Francoise gave an algorithm for calculating the first Nonzero Term of the displacement function Δ along γ of such deformations. This algorithm recalls the well-known Godbillon–Vey sequences discovered in 1971 for investigation of integrability of a form ω. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon–Vey sequences to the Francoise algorithm settings.
Jessie Pontigoherrera - One of the best experts on this subject based on the ideXlab platform.
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bounding the length of iterated integrals of the first Nonzero melnikov function
Moscow Mathematical Journal, 2018Co-Authors: Pavao Mardesic, Dmitry Novikov, Laura Ortizbobadilla, Jessie PontigoherreraAbstract:We consider small polynomial deformations of integrable systems of the form $dF=0, F\in\mathbb{C}[x,y]$ and the first Nonzero Term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ epends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first Nonzero Term $\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.
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bounding the length of iterated integrals of the first Nonzero melnikov function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Pavao Mardesic, Dmitry Novikov, Laura Ortizbobadilla, Jessie PontigoherreraAbstract:We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb{C}[x,y]$ and the first Nonzero Term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ depends on the deformation of $dF$. In this paper we give a universal bound for the length of the iterated integral expressing the first Nonzero Term $M_\mu$ depending only on the topology of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral i.e. by an iterated integral of length $1$. We conjecture that our bound is optimal.
