Hamiltonian Approximation Research Articles

The Hall conductivity of graphene

Sensitivity of the Hall conductivity of graphene plane to gas molecule adsorption is investigated within the coherent potential approximation for the tight-binding model Hamiltonian. The results show that the Hall conductivity of system have a limit change when finite triatomic and tetratomic gas molecules adsorb and act as acceptors or donors.

Accurate calculations on the 12 electronic states and 23 Ω states of the SiBr+ cation: potential energy curves, spectroscopic parameters and spin-orbit coupling

The potential energy curves (PECs) of 23 Ω states generated from the 12 electronic states (X1 Σ +, 21 Σ +, 11 Σ −, 11 Π, 21 Π, 11 Δ, 13 Σ +, 23 Σ +, 13 Σ −, a3 Π, 23 Π and 13 Δ) are studied for the first time. All the states correlate to the first dissociation channel of the SiBr+ cation. Of these electronic states, the 23 Σ + is the repulsive one without the spin-orbit coupling, whereas it becomes the bound one with the spin-orbit coupling added. On the one hand, without the spin-orbit coupling, the 11 Π, 21 Π and 23 Π are the rather weakly bound states, and only the 11 Π state possesses the double well; on the other hand, with the spin-orbit coupling included, the a3 Π and 11 Π states possess the double well, and the 13 Σ + and 13 Σ − are the inverted states. The PECs are calculated by the CASSCF method, which is followed by the internally contracted MRCI approach with the Davidson modification. Scalar relativistic correction is calculated by the third-order Douglas-Kroll Hamiltonian approximation with a cc-pVTZ-DK basis set. Core-valence correlation correction is included with a cc-pCVTZ basis set. The spin-orbit coupling is accounted for by the state interaction method with the Breit-Pauli Hamiltonian using the all-electron aug-cc-pCVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The variation with internuclear separation of the spin-orbit coupling constant is discussed in brief. The spectroscopic parameters are evaluated for the 11 bound electronic states and the 23 bound Ω states, and are compared with available measurements. Excellent agreement has been found between the present results and the experimental data. It demonstrates that the spectroscopic parameters reported here can be expected to be reliably predicted ones. The Franck-Condon factors and radiative lifetimes of the transitions from the a3 Π 0 + and a3 Π 1 states to the X1 Σ + 0+ state are calculated for several low vibrational levels, and some brief discussion has been made.

Extensive spectroscopic calculations on 12 low-lying electronic states of AlN molecule including transition properties

Using the CASSCF method followed by the internally contracted MRCI approach in combination with the correlation-consistent basis sets, the potential energy curves (PECs) are calculated for the X3Π, A3Σ-, B3Σ+, C3Π, E3Δ, a1Σ+, b1Π, c1Δ, d1Σ+, e1Π, 23Σ− and 33Σ− electronic states of AlN molecule for internuclear separations from 0.1 to 1.0 nm. All the electronic states correlate to the three dissociation channels, Al(2Pu) + N(4Su), Al(2Pu) + N(2Du) and Al(2Pu) + N(2Pu). Of these 12 electronic states, only the 23Σ− possesses the double well. The PECs determined by the internally contracted MRCI approach are corrected for size-extensivity errors by means of the Davidson correction. The convergent behavior of present calculations is observed with respect to the basis set and level of theory. The effect of core-valence correlation and scalar relativistic corrections on the spectroscopic parameters is discussed. Scalar relativistic correction calculations are performed by the third-order Douglas-Kroll Hamiltonian approximation at the level of cc-pVTZ basis set. Core-valence correlation corrections are included with a cc-pCVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are evaluated by fitting the first ten vibrational levels when available, which are obtained by solving the ro-vibrational Schrodinger equation with the Numerov’s method. The spectroscopic parameters are compared with those reported in the literature. Excellent agreement is found between the present results and the measurements. Analyses show that the spectroscopic parameters reported in this paper can be expected to be reliably predicted ones. The Franck-Condon factors and radiative lifetimes of the transitions from the A3Σ−, B3Σ+, C3Π, a1Σ+ and b1Π electronic states to the ground state are calculated for several low vibrational levels, and some necessary discussion has been made.

Numerical approximation of the Schrödinger equation with the electromagnetic field by the Hagedorn wave packets

In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error O(ε1/2), where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to O(εk/2), where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small.

The influence of a presence of a heavy atom on the spin-spin coupling constants between two light nuclei in organometallic compounds and halogen derivatives

The (1)JCC and (1)JCH spin-spin coupling constants have been calculated by means of density functional theory (DFT) for a set of derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg in order to evaluate the substituent and relativistic effects for these properties. The main goal was to estimate HALA (heavy-atom-on-light-atom) effects on spin-spin coupling constants and to explore the factors which may influence the HALA effect on these properties, including the nature of the heavy atom substituent and carbon hybridization. The methods applied range, in order of reduced complexity, from Dirac-Kohn-Sham method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component Zeroth Order Regular Approximation (ZORA) Hamiltonians, to scalar non-relativistic effective core potentials with the non-relativistic Hamiltonian. Thus, we are able to compare the performance of ZORA-DFT and Dirac-Kohn-Sham methods for modelling of the HALA effects on the spin-spin coupling constants.

Nematic phase in stripe-forming systems within the self-consistent screening approximation

We show that in order to describe the isotropic-nematic transition in stripe-forming systems with isotropic competing interactions of the Brazovskii class it is necessary to consider the next to leading order in a 1/N approximation for the effective Hamiltonian. This can be conveniently accomplished within the self-consistent screening approximation. We solve the relevant equations and show that the self-energy in this approximation is able to generate the essential wave vector dependence to account for the anisotropic character of a two-point correlation function characteristic of a nematic phase.

Perspective: Relativistic Hamiltonians

Under the no‐photon approximation, an effective quantum electrodynamics (QED) Hamiltonian can be constructed based solely on charge conjugation symmetry. It serves as the basis not only for the emerging field of “molecular QED” but also for various approximate Hamiltonians, including the standard no‐pair ones. The whole spectrum of relativistic Hamiltonians is hence completed. © 2013 Wiley Periodicals, Inc.

Accurate calculations on 12 Λ–S and 28 Ω states of BN+ cation: Potential energy curves, spectroscopic parameters and spin–orbit coupling

The potential energy curves (PECs) of 28Ω states generated from the 12 states (X4Σ−, 12Π, 12Σ−, 12Δ, 12Σ+, 22Π, A4Π, B4Σ−, 32Π, 16Σ−, 22Σ− and 16Π) of the BN+ cation are studied for the first time for internuclear separations from about 0.1 to 1.0nm using an ab initio quantum chemical method. All the Λ–S states correlate to the first four dissociation channels. The 16Σ−, 32Π and A4Π states are found to be the inverted ones. The 12Σ+, 22Π, 32Π and 22Σ− states are found to possess the double well. The PECs are calculated by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson correction. Core–valence correlation correction is included by a cc-pCV5Z basis set. Scalar relativistic correction is calculated by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. The convergent behavior of present calculations is discussed with respect to the basis set and level of theory. The spin–orbit coupling is accounted for by the state interaction approach with the Breit–Pauli Hamiltonian using the all-electron cc-pCV5Z basis set. All the PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are obtained, and the vibrational properties of 12Σ+, 22Π, 32Π and 22Σ− states are evaluated. Analyses demonstrate that the spectroscopic parameters reported here can be expected to be reliably predicted ones. The conclusion is gained that the effect of spin–orbit coupling on the spectroscopic parameters are not obvious almost for all the Λ–S states involved in the present paper.

Theoretical studies on 14 Λ-S and 30 Ω states of BeBr molecule: potential energy curves, spectroscopic parameters and spin–orbit couplings

Using the complete active space self-consistent field (CASSCF) method followed by the internally contracted multi-reference configuration interaction (MRCI) approach in combination with the correlation-consistent basis sets, this paper studies the potential energy curves of X2Σ+, 22Σ+, 32Σ+, 12Σ−, A2Π, 22Π, 32Π, 12Δ, 14Σ+, 24Σ+, 14Σ−, 14Π, 24Π and 14Δ Λ-S states of BeBr molecule and the corresponding 30 Ω states for the first time. All the Λ-S states correlate to the first two dissociation channels, Be(1Sg) + Br(2Pu) and Be(3Pu) + Br(2Pu), of BeBr molecule. Of these Λ-S states, the 32Π and 24Π are found to be repulsive without the spin–orbit coupling, whereas 14Π, 24Π, 32Π and 24Σ+ are found to be repulsive with the spin–orbit coupling included. A2Π and 22Σ+ possess the double well whether the spin–orbit coupling effect is included or not. Only 14Σ+, 14Σ−, 12Π and 22Π are found to be the inverted Λ-S states. The spin–orbit coupling is accounted for by the state interaction approach with Breit–Pauli Hamiltonian using the all-electron cc-pCVTZ basis set. The potential energy curves determined by the internally contracted MRCI method are corrected for size-extensivity errors by means of the Davidson correction. Core–valence correlation correction is calculated with a cc-pCVTZ basis set. Scalar relativistic correction is included using the third-order Douglas–Kroll Hamiltonian approximation at the level of cc-pVTZ basis set. The spectroscopic parameters of all the Λ-S and Ω bound states are evaluated. The spectroscopic parameters are compared with those reported in the literature. Fair agreement is found between the present results and available measurements. In particular, the energy splitting of 204.43 cm−1 in the A2Π Λ-S state agrees well with the measurements of 201 cm−1. Analyses demonstrate that the spectroscopic parameters reported here can be expected to be reliably predicted ones.

Theoretical study on thirteen Λ-S states of the SiN radical: potential energy curves, spectroscopic parameters and spin-orbit couplings

The spectroscopic properties of nineteen Ω states generated from the eight Λ-S bound states of SiN radicals are studied in detail using an ab initio quantum chemical method for the first time. The potential energy curves (PECs) are calculated by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification (icMRCI+Q). The spin-orbit coupling is accounted for by the Breit-Pauli Hamiltonian using the cc-pCVTZ basis set. Scalar relativistic correction calculations are made using the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Core-valence correlation corrections are included with a cc-pCVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The convergent behavior is discussed with respect to the basis set and level of theory. The effect of core-electron correlations on the energy splitting is studied. Using the PECs, the spectroscopic parameters of twelve Λ-S and nineteen Ω bound states are evaluated. At the icMRCI+Q/CV+DK+56 theoretical level, the spin-orbit coupling splitting energy of the A 2 ΠΛ-S state is found to be 88.45 cm-1, which agrees well with the recent measurements of 89.09 cm-1. Moreover, other spectroscopic parameters are also in fair agreement with their respective measurements. The effect of SO coupling on the vibrational levels is discussed. It shows that the spectroscopic parameters of the twelve Λ-S and nineteen Ω bound states reported here can be expected to be reliably predicted ones.

Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration

We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations with no restoring force. These rotations determine states in the Goldstone manifold, which originates from a spontaneously broken continuous symmetry in the wave function. Zero modes can be classified as improper or proper according to their different mathematical and physical properties. Improper modes arise from symmetry breaking and their restoration always lowers the energy. Proper modes, on the other hand, correspond to degeneracies of the wave function, and their symmetry restoration does not necessarily lower the energy. We discuss how the RPA Hamiltonian distinguishes between proper and improper modes by doubling the number of zero eigenvalues associated with the latter. Proper modes in the Hessian always appear in pairs which do not double in RPA. We present several pedagogical cases exemplifying the above statements. The relevance of these results for projected Hartree-Fock methods is also addressed.

Extensive calculations on 12 Λ-S and 27 Ω states of PCl+ cation including spin–orbit coupling

The potential energy curves (PECs) of 27 Ω states generated from the 12 Λ-S states (X2Π, A2Π, 14Π, 24Π, 12Σ−, 22Σ−, 14Σ−, 24Σ−, 12Σ+, 14Σ+, 12Δ and 14Δ) of PCl+ cation are studied for the first time for internuclear separations from about 0.10 to 1.10nm using an ab initio quantum chemical method. All the 12 Λ-S states correlate to the first dissociation channel of PCl+ cation. Of these Λ-S states, the 24Π is found to be the repulsive one. The 14Σ+, 12Δ and 14Δ are found to be the inverted ones. And the 12Δ is found to possess the double wells. The PECs are calculated by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson correction in combination with the correlation-consistent basis sets, aug-cc-pV(n+d)Z. The effect of core–valence correlation and scalar relativistic corrections on the spectroscopic parameters is briefly discussed. Scalar relativistic corrections are included by the third-order Douglas–Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Core–valence correlation corrections are included with a cc-pCVTZ basis set. The convergent behavior of present calculations is discussed with respect to the basis set and level of theory. The spin–orbit coupling is accounted for by the state interaction method with the Breit–Pauli Hamiltonian using the all-electron cc-pCVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are evaluated for the 11 Λ-S bound states and for the 23 Ω bound states, and are compared with available experimental and other theoretical results. Fair agreement has been found between the present spectroscopic parameters and the measurements. The energy splitting in the X2Π Λ-S state is calculated to be 346.11cm−1, close to the estimated measurements of 370cm−1. It demonstrates that the spectroscopic parameters reported here can be expected to be reliably predicted ones.

Theoretical calculations on 12 Λ-S and 23 Ω states of CBr+ cation in the gas phase: Potential energy curves, spectroscopic parameters and spin–orbit coupling

The potential energy curves (PECs) of X1Σ+, a3Π, 13Σ+, 13Δ, 11Δ, 11Σ−, 13Σ−, 11Π, 21Σ+, 23Π, 21Π and 23Σ+ Λ-S states of CBr+ cation and corresponding 23 Ω states are calculated for the first time using the CASSCF method, which is followed by the internally contracted MRCI approach with the aug-cc-pVQZ basis set. All the Λ-S states involved are found to be bound and dissociate into the first dissociation limit of CBr+ cation. Of these Λ-S states, only the 13Σ+ and 13Σ− are inverted ones. The spin–orbit (SO) coupling is accounted for by the state interaction approach with the Breit–Pauli Hamiltonian. Core–valence correlation is included by a cc-pCVTZ basis set. Relativistic correction is calculated with the third-order Douglas–Kroll Hamiltonian approximation at the level of cc-pVQZ basis set. To obtain more reliable results, the PECs obtained by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification. The PEC crossings of different Λ-S states are studied. With these PECs, the spectroscopic parameters of all the Λ-S and Ω states involved are obtained by fitting the first ten vibrational levels whenever available, which are calculated by solving the rovibrational Schrödinger equation using the Numerov's method. The spectroscopic parameters are compared with those reported in the literature. Excellent agreement is found between the present results and available measurements. In particular, the energy separation of 352.26cm−1 between the a3Π0+ and the a3Π1 Ω states agrees well with the measurements of 369±8cm−1, and the ωe results of 907.45 and 907.08cm−1 for the a3Π0+ and a3Π1 Ω states are in excellent agreement with the measurements of 906±2 and 903±6cm−1, respectively. These show that the spectroscopic parameters obtained in the present paper can be expected to be reliable predicted ones.

The influence of the zigzag and armchair leads on quantum transport through the graphene based quantum rings

The conductivity of a graphene ring with two semi-infinite, armchair and zigzag leads has been investigated. We have performed numerical calculations based on the nearest neighbor tight-binding Hamiltonian and Dirac point approximation. A Non-Equilibrium Green's Function (NEGF) approach has been employed to calculate the electric current under an applied bias voltage, in this two terminal mesoscopic system. We have studied the effect of the external magnetic field on transport characteristics of this graphene-based quantum ring. Coherent transport features of the system have been studied. It was shown that there is significant distinction between the I---V characteristics (and also the Aharonov-Bohm effect) of the graphene rings depending on the edge structure of the leads.

Effective Hamiltonians for Complexes of Unstable Particles

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Królikowski–Rzewuski (KR) equation. KR equation results from the Schrödinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for n-particle, two-particle and for one-particle subspaces. In a general case these effective Hamiltonians depend on time t. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one-particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when t → ∞ whereas the imaginary part of this effective Hamiltonian tends to zero as t → ∞.

Multireference configuration interaction study on spectroscopic properties of low-lying electronic states of As2 molecule

The potential energy curves (PECs) of four electronic states (X1Σg+, e3Δu, a3Σu−, and d3Πg) of an As2 molecule are investigated employing the complete active space self-consistent field (CASSCF) method followed by the valence internally contracted multireference configuration interaction (MRCI) approach in conjunction with the correlation-consistent aug-cc-pV5Z basis set. The effect on PECs by the relativistic correction is taken into account. The way to consider the relativistic correction is to employ the second-order Douglas-Kroll Hamiltonian approximation. The correction is made at the level of a cc-pV5Z basis set. The PECs of the electronic states involved are extrapolated to the complete basis set limit. With the PECs, the spectroscopic parameters (Te, Re, ωe, ωexe, ωeye, αe, βe, γe, and Be) of these electronic states are determined and compared in detail with those reported in the literature. Excellent agreement is found between the present results and the experimental data. The first 40 vibrational states are studied for each electronic state when the rotational quantum number J equals zero. In addition, the vibrational levels, inertial rotation and centrifugal distortion constants of d3Πg electronic state are reported which are in excellent agreement with the available measurements. Comparison with the experimental data shows that the present results are both reliable and accurate.

Accurate calculations on 9 Λ-S and 28 Ω states of NSe radical in the gas phase: Potential energy curves, spectroscopic parameters and spin–orbit couplings

The potential energy curves (PECs) of 28 Ω states generated from 9 Λ-S states (X2Π, 14Π, 16Π, 12Σ+, 14Σ+, 16Σ+, 14Σ-, 24Π and 14Δ) are studied for the first time using an ab initio quantum chemical method. All the 9 Λ-S states correlate to the first two dissociation limits, N(4Su)+Se(3Pg) and N(4Su)+Se(3Dg), of NSe radical. Of these Λ-S states, the 16Σ+, 14Σ+, 16Π, 24Π and 14Δ are found to be rather weakly bound states. The 12Σ+ is found to be unstable and has double wells. And the 16Σ+, 14Σ+, 14Π and 16Π are found to be the inverted ones with the SO coupling included. The PEC calculations are made by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification. The spin–orbit coupling is accounted for by the state interaction approach with the Breit-Pauli Hamiltonian. The convergence of the present calculations is discussed with respect to the basis set and the level of theory. Core-valence correlation corrections are included with a cc-pCVTZ basis set. Scalar relativistic corrections are calculated by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. All the PECs are extrapolated to the complete basis set limit. The variation with internuclear separation of spin–orbit coupling constants is discussed in brief for some Λ-S states with one shallow well on each PEC. The spectroscopic parameters of 9 Λ-S and 28 Ω states are determined by fitting the first ten vibrational levels whenever available, which are calculated by solving the rovibrational Schrödinger equation with Numerov’s method. The splitting energy in the X2Π Λ-S state is determined to be about 864.92cm−1, which agrees favorably with the measurements of 891.80cm−1. Moreover, other spectroscopic parameters of Λ-S and Ω states involved here are also in fair agreement with available measurements. It demonstrates that the spectroscopic parameters reported here can be expected to be reliable predicted ones.

Accurate theoretical study on 18 Λ-S and 50 Ω states of CS in the gas phase: Potential energy curves, spectroscopic parameters, and spin-orbit coupling

The potential energy curves (PECs) of 50 Ω states generated from the 18 Λ-S states are studied for the first time using the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification. All the 18 Λ-S states correlate to the first dissociation limit, C((3)Pg) + S((3)Pg), of CS molecule, of which only the 2(5)Π is repulsive and the A(1)Π, A(') (1)Σ(+), and 2(3)Σ(+) possess double wells. The spin-orbit (SO) coupling is accounted for by the state interaction approach with the Breit-Pauli Hamiltonian. Core-valence correlation correction is taken into account with an all-electron cc-pCV5Z basis set. Scalar relativistic correction calculations are made by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The a(') (3)Σ(+), e(3)Σ(-), 1(5)Σ(+), 1(5)Π, and d(3)Δ are found to be the inverted Λ-S states with the SO coupling included. The spectroscopic parameters of 17 Λ-S and 41 Ω bound states are evaluated. The comparisons between the present results and available measurements are performed, and excellent agreement has been found. It shows that the spectroscopic results reported here can be expected to be reliable predicted ones.

SYMMETRY PROPERTIES OF THE EXACT EM RADIATION-REACTION EQUATION FOR CLASSICAL EXTENDED PARTICLES AND ANTIPARTICLES

Based on recent theoretical developments (Cremaschini and Tessarotto, 2011–2013), in this paper the issue is addressed of the first-principle construction of the nonlocal relativistic radiation-reaction (RR) equation for classical spherical-shell, finite-size particles and antiparticles. This is achieved invoking the axioms of Classical Electrodynamics by means of the Hamilton variational principle. In connection with this, the Lagrangian conservation laws, together with the possible existence of adiabatic invariants, and the transformation laws of the RR equation with respect to CPT and time-reversal transformations are investigated. The latter properties make possible the parametrization of the RR equations, holding respectively for particles and antiparticles of this type, in terms of the same coordinate time t and the investigation of the qualitative properties of their solutions. In particular, in both cases the RR self-force is found to have the same signature, which implies that the dynamics of classical finite-size antiparticles is equivalent to that of classical extended particles of opposite charge. Therefore, in the framework of Classical Mechanics, a distinction between particles and antiparticles cannot be made based solely on the electromagnetic interactions associated with electromagnetic RR phenomena. As a basic application of the theory, the Lagrangian conservation laws and symmetry properties for the Hamiltonian asymptotic approximations of the exact RR equation are also addressed.

On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev

The application of the Nekhoroshev theorem to many problems arising in different fields of Physics and Astronomy depends on a non-degeneracy property, called steepness, that a suitable Hamiltonian approximation must satisfy. Since steepness is implicitly defined, we have the problem of recognizing whether a given function is steep or not. For this purpose, we here consider some sufficient conditions for steepness provided by Nekhoroshev in 1979, based on the solvability of a collection of systems depending on the number n of degrees of freedom, the derivatives of the function up to a certain order r, and some auxiliary parameters. These conditions are really explicit only for r = 2, corresponding to quasi-convexity , and for r = 3. Instead, for r ⩾ 4, the conditions are implicit, since they require an elaborate computation of the closure of a certain set. In this paper, we first revisit Nekhoroshev's result and we show that the number of parameters in the collections of systems can be suitably reduced. Then, we show that for r = 4 Nekhoroshev's result is interesting only for n = 2, 3, and 4, and in these cases we find explicit conditions for steepness which are formulated in a purely algebraic form.